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1 
Topics in torsion theoryCharalambides, Stelios, n/a January 2006 (has links)
The purpose of this thesis is to generalize to the torsiontheoretic setting various concepts and results from the theory of rings and modules. In order to accomplish this we begin with some preliminaries which introduce the main ideas used in torsion theory, the major ones being [tau]torsion and [tau]torsionfree modules as well as [tau]dense and [tau]pure submodules. In the first chapter we also introduce a new concept, that of a [tau]compact module, which is basic enough to deserve a place among the preliminaries.
The results that we obtain fall into three areas which are to a certain degree interrelated. The first area is on [tau]Max modules, which we introduce as a torsiontheoretic analogue of Max modules. The main aim is to generalize a wellknown result by Shock which characterizes Noetherian rings by using the socle, the radical and Max modules. All of these concepts have torsiontheoretic counterparts which we utilize in our generalization. Furthermore, we define and characterize left [tau]Max rings and apply the torsiontheoretic version of Shock�s theorem to obtain a characterization of [tau]short modules motivated by a recent article in which short modules were introduced.
The second area deals with various flavours of [tau]injectivity, some known and some new. We introduce [tau]Minjective and s[tau]Minjective modules and examine their relationship with the known concepts of [tau]injective and [tau]quasiinjective modules. We then provide an improved version of the Generalized Fuchs Criterion which characterizes s[tau]Minjective modules, and give a generalization of Azumaya�s Lemma. We also prove that every Mgenerated module has a [tau]Minjective hull which is unique up to isomorphism and show how this is linked to the [tau]quasiinjective hull. We then examine [Sigma][tau]injectivity, generalizing wellknown results by Faith, Albu and Năstăsescu and Cailleau which provide necessary and sufficient conditions for the [Sigma][tau]injective property, the [Sigma]s[tau]Minjective property and for a direct sum of [Sigma]s[tau]Minjective modules to be [Sigma]s[tau]Minjective.
In the third area we introduce a couple of new concepts with the aim of bringing to the torsiontheoretic setting the concept of a CS or extending module. The approach is twofold. The first is via [tau]CS modules which serve as a generalization of CS modules as well as [tau]quasicontinuous, [tau]quasiinjective and [tau]injective modules, and the second is via s[tau]CS modules which are a special case of CS modules. Our motivation is to provide a torsiontheoretic analogue of a wellknown result by Okado which characterizes Noetherian modules. We have some partial results using s[tau]CS modules and a nice torsiontheoretic analogue, albeit without the use of [tau]CS or s[tau]CS modules. We also examine the relationship between our relative versions of CS modules with those of other authors and obtain refinements to some of their results.

2 
Topics in torsion theoryCharalambides, Stelios, n/a January 2006 (has links)
The purpose of this thesis is to generalize to the torsiontheoretic setting various concepts and results from the theory of rings and modules. In order to accomplish this we begin with some preliminaries which introduce the main ideas used in torsion theory, the major ones being [tau]torsion and [tau]torsionfree modules as well as [tau]dense and [tau]pure submodules. In the first chapter we also introduce a new concept, that of a [tau]compact module, which is basic enough to deserve a place among the preliminaries.
The results that we obtain fall into three areas which are to a certain degree interrelated. The first area is on [tau]Max modules, which we introduce as a torsiontheoretic analogue of Max modules. The main aim is to generalize a wellknown result by Shock which characterizes Noetherian rings by using the socle, the radical and Max modules. All of these concepts have torsiontheoretic counterparts which we utilize in our generalization. Furthermore, we define and characterize left [tau]Max rings and apply the torsiontheoretic version of Shock�s theorem to obtain a characterization of [tau]short modules motivated by a recent article in which short modules were introduced.
The second area deals with various flavours of [tau]injectivity, some known and some new. We introduce [tau]Minjective and s[tau]Minjective modules and examine their relationship with the known concepts of [tau]injective and [tau]quasiinjective modules. We then provide an improved version of the Generalized Fuchs Criterion which characterizes s[tau]Minjective modules, and give a generalization of Azumaya�s Lemma. We also prove that every Mgenerated module has a [tau]Minjective hull which is unique up to isomorphism and show how this is linked to the [tau]quasiinjective hull. We then examine [Sigma][tau]injectivity, generalizing wellknown results by Faith, Albu and Năstăsescu and Cailleau which provide necessary and sufficient conditions for the [Sigma][tau]injective property, the [Sigma]s[tau]Minjective property and for a direct sum of [Sigma]s[tau]Minjective modules to be [Sigma]s[tau]Minjective.
In the third area we introduce a couple of new concepts with the aim of bringing to the torsiontheoretic setting the concept of a CS or extending module. The approach is twofold. The first is via [tau]CS modules which serve as a generalization of CS modules as well as [tau]quasicontinuous, [tau]quasiinjective and [tau]injective modules, and the second is via s[tau]CS modules which are a special case of CS modules. Our motivation is to provide a torsiontheoretic analogue of a wellknown result by Okado which characterizes Noetherian modules. We have some partial results using s[tau]CS modules and a nice torsiontheoretic analogue, albeit without the use of [tau]CS or s[tau]CS modules. We also examine the relationship between our relative versions of CS modules with those of other authors and obtain refinements to some of their results.

3 
FIELDS DEFINED BY RADICALS: THEIR TORSION GROUP AND THEIR LATTICE OF SUBFIELDS.ACOSTA DE OROZCO, MARIA TEODORA. January 1987 (has links)
Let L/F be a finite separable extension. L* = L\{0}, and T(L*/F*) be the torsion subgroup of L*/F*. We explicitly determined T(L*/F*) when L/F is an abelian extension. This information is used to study the structure of T(L*/F*). In particular T(F(α)*/F*) when αᵐ = a ∈ F is explicitly determined. Let Xᵐ  a be irreducible over F with char F χ m and let α be a root of Xᵐ  a. We study the lattice of subfields of F(α)/F and to this end C(F(α)/F,k) is defined to be the number of subfields of F(α) of degree k over F. C(f(α)/F,pⁿ) is explicitly determined for p a prime and the following structure theorem for the lattice of subfields is proved. Let N be the maximal normal subfield of F(α) and set n = [N:F], then C(F(α)/F,k) = C(F(α)/F,(k,n)) = C(N/F,(k,n)). The irreducible binomials X⁸  b, X⁸  c are said be equivalent if there exist roots β⁸ = b, γ⁸ = c that F(β) = F(γ). All the mutually inequivalent binomials which have roots in F(α) are determined. These results are applied the study of normal binomials and those irreducible binomials X²ᵉ  a which are normal over F(charF ≠ 2) together their Galois groups are characterized. We finished by considering the radical extension F(α)/F, αᵐ ∈ F, where the binominal Xᵐ  αᵐ is not necessarily irreducible. We see that in the case not every subfield of F(α)/F is the compositum of subfields of prime power order. We determine some conditions such that if F ⊆ H ⊆ F(α) with [H:F] = pᵘq, p a prime, (p,q) = 1, then there exists a subfield F ⊆ R ⊆ H where [R:F] = pᵘ.

4 
Torsion theories and localizations for MsetsGuruswami, Verena January 1976 (has links)
No description available.

5 
Torsion in the homology of the general linear group for a ring of algebraic integers /Adhikari, S. Prashanth, January 1997 (has links)
Thesis (Ph. D.)University of Washington, 1997. / Vita. Includes bibliographical references (leaves [117]121).

6 
Zur Torsion der Kohomologie Sarithmetischer GruppenHesselmann, Sabine. January 1993 (has links)
Thesis (doctoral)Rheinische FriedrichWilhelmsUniversität Bonn, 1992. / Includes bibliographical references (p. 9193).

7 
Torsion theories and localizations for MsetsGuruswami, Verena January 1976 (has links)
No description available.

8 
On purity relative to an hereditary torsion theory.Gray, Derek Johanathan. January 1992 (has links)
The thesis is mainly concerned with properties of the concept
"σpurity" introduced by J. Lambek in "Torsion Theories, Additive
Semantics and Rings of Quotients", (SpringerVerlag, 1971).
In particular we are interested in modul es M for which every exact
sequence of the form O→M→K→L→O (or O→K→M→L→O or O→K→L→M→O) is σpure
exact. Modules of the first type turn out to be precisely the
σ injective modules of O. Goldman (J. Algebra 13, (1969), 1047).
This characterization allows us to study σ injectivity from the perspective of purity.
Similarly the demand that every short exact sequence of modules of the form O→K→M→L→O or O→K→L→M→O be σpure exact leads to concepts which generalize regularity and flatness respectively. The questions of which properties of regularity and flatness extend to these more general concepts of σ regularity and σflatness are investigated.
For various classes of rings R and torsion radicals σ on Rmod, certain conditions equivalent to the σregularity and the σinjectivity of R are found.
We also introduce some new dimensions and study semiσflat and
semiσinjective modules (defined by suitably restricting conditions
on σflat and σinjective modules). We further characterize those rings R for which every Rmodule is semi σflat.
The related concepts of a projective cover and a perfect ring
(introduced by H. Bass in Trans. Amer. Math. Soc. 95, (1960), 466488)
are extended in a 'natural way and, inter alia , we obtain a generalization of a famous theorem of Bass.
Lastly, we develop a relativized version of the Jacobson Radical which is shown to have properties analogous to both the classical Jacobson Radical and a radical due to J.S. Golan. / Thesis (Ph.D.)University of Natal, 1992.

9 
Torsion points on elliptic curvesNyirenda, Darlison 03 1900 (has links)
Thesis (MSc)Stellenbosch University, 2013. / ENGLISH ABSTRACT: The central objective of our study focuses on torsion points on elliptic curves. The case of
elliptic curves over finite fields is explored up to giving explicit formulae for the cardinality
of the set of points on such curves. For finitely generated fields of characteristic zero, a
presentation and discussion of some known results is made. Some applications of elliptic
curves are provided. In one particular case of applications, we implement an integer
factorization algorithm in a computer algebra system SAGE based on Lenstra’s elliptic
curve factorisation method. / AFRIKAANSE OPSOMMING: Die hoofdoel van ons studie is torsiepunte op elliptiese krommes. Ons ondersoek die geval
van elliptiese krommes oor ‘n eindige liggaam met die doel om eksplisiete formules vir die
aantal punte op sulke krommes te gee. Vir ‘n eindigvoortgebringde liggaam met karakteristiek
nul bespreek ons sekere bekende resultate. Sommige toepassings van elliptiese
krommes word gegee. In een van hierdie toepassings implementeer ons ‘n heeltallige faktoriseringalgoritme
in die rekenaaralgebrastelsel SAGE gebaseer op Lenstra se elliptiese
krommefaktoriseeringmetode.

10 
Ore localizations and irreducible representations of the first Weyl algebra.Zhang, YingLan. Muller, Bruno, J. Unknown Date (has links)
Thesis (Ph.D.)McMaster University (Canada), 1990. / Source: Dissertation Abstracts International, Volume: 5210, Section: B, page: 5315. Supervisor: Bruno J. Muller.

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