The purpose of this thesis is to generalize to the torsion-theoretic 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 torsion-theoretic analogue of Max modules. The main aim is to generalize a well-known result by Shock which characterizes Noetherian rings by using the socle, the radical and Max modules. All of these concepts have torsion-theoretic counterparts which we utilize in our generalization. Furthermore, we define and characterize left [tau]-Max rings and apply the torsion-theoretic 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]-M-injective and s-[tau]-M-injective modules and examine their relationship with the known concepts of [tau]-injective and [tau]-quasi-injective modules. We then provide an improved version of the Generalized Fuchs Criterion which characterizes s-[tau]-M-injective modules, and give a generalization of Azumaya�s Lemma. We also prove that every M-generated module has a [tau]-M-injective hull which is unique up to isomorphism and show how this is linked to the [tau]-quasi-injective hull. We then examine [Sigma]-[tau]-injectivity, generalizing well-known results by Faith, Albu and Năstăsescu and Cailleau which provide necessary and sufficient conditions for the [Sigma]-[tau]-injective property, the [Sigma]-s-[tau]-M-injective property and for a direct sum of [Sigma]-s-[tau]-M-injective modules to be [Sigma]-s-[tau]-M-injective. In the third area we introduce a couple of new concepts with the aim of bringing to the torsion-theoretic 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]-quasi-continuous, [tau]-quasi-injective 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 torsion-theoretic analogue of a well-known result by Okado which characterizes Noetherian modules. We have some partial results using s-[tau]-CS modules and a nice torsion-theoretic 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.

The purpose of this thesis is to generalize to the torsion-theoretic 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 torsion-theoretic analogue of Max modules. The main aim is to generalize a well-known result by Shock which characterizes Noetherian rings by using the socle, the radical and Max modules. All of these concepts have torsion-theoretic counterparts which we utilize in our generalization. Furthermore, we define and characterize left [tau]-Max rings and apply the torsion-theoretic 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]-M-injective and s-[tau]-M-injective modules and examine their relationship with the known concepts of [tau]-injective and [tau]-quasi-injective modules. We then provide an improved version of the Generalized Fuchs Criterion which characterizes s-[tau]-M-injective modules, and give a generalization of Azumaya�s Lemma. We also prove that every M-generated module has a [tau]-M-injective hull which is unique up to isomorphism and show how this is linked to the [tau]-quasi-injective hull. We then examine [Sigma]-[tau]-injectivity, generalizing well-known results by Faith, Albu and Năstăsescu and Cailleau which provide necessary and sufficient conditions for the [Sigma]-[tau]-injective property, the [Sigma]-s-[tau]-M-injective property and for a direct sum of [Sigma]-s-[tau]-M-injective modules to be [Sigma]-s-[tau]-M-injective. In the third area we introduce a couple of new concepts with the aim of bringing to the torsion-theoretic 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]-quasi-continuous, [tau]-quasi-injective 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 torsion-theoretic analogue of a well-known result by Okado which characterizes Noetherian modules. We have some partial results using s-[tau]-CS modules and a nice torsion-theoretic 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.

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.

No description available.

Thesis (Ph. D.)--University of Washington, 1997. / Vita. Includes bibliographical references (leaves [117]-121).

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1992. / Includes bibliographical references (p. 91-93).

No description available.

The thesis is mainly concerned with properties of the concept "σ-purity" introduced by J. Lambek in "Torsion Theories, Additive Semantics and Rings of Quotients", (Springer-Verlag, 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 σ-pure exact. Modules of the first type turn out to be precisely the σ- injective modules of O. Goldman (J. Algebra 13, (1969), 10-47). 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 σ-pure 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 σ-flatness are investigated. For various classes of rings R and torsion radicals σ on R-mod, certain conditions equivalent to the σ-regularity and the σ-injectivity of R are found. We also introduce some new dimensions and study semi-σ-flat and semi-σ-injective modules (defined by suitably restricting conditions on σ-flat and σ-injective modules). We further characterize those rings R for which every R-module is semi- σ-flat. The related concepts of a projective cover and a perfect ring (introduced by H. Bass in Trans. Amer. Math. Soc. 95, (1960), 466-488) 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.

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 eindig-voortgebringde 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 rekenaar-algebrastelsel SAGE gebaseer op Lenstra se elliptiese krommefaktoriseeringmetode.

Thesis (Ph.D.)--McMaster University (Canada), 1990. / Source: Dissertation Abstracts International, Volume: 52-10, Section: B, page: 5315. Supervisor: Bruno J. Muller.