Thesis (M.S.)--Missouri State University, 2008. / "August 2008." Includes bibliographical references (leaf 51). Also available online.

Thesis (M.S.)--Missouri State University, 2008. / "August 2008." Includes bibliographical references (leaf 53). Also available online.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2006. / Title from title screen (sites viewed on August 10, 2006). PDF text of dissertation: 86 p. ; 0.41Mb. UMI publication number: AAT 3208054. Includes bibliographical references. Also available in microfilm, microfiche and paper format.

Fix R = k[[x]]. Let Q_n be the category whose objects are ((M₁,...,M_n),(f₁,...,f_n-1)) where each M_i is a free R-module and f_i:M_i⟶M_i+1 for each i=1,...,n-1, and in which the morphisms are the obvious ones. Let β_n be the full subcategory of Ω_n in which each map f_i is a monomorphism whose cokernel is a torsion module. It is shown that there is a full dense functor Ω_n⟶β_n. If X is an object of β_n, we say that X <u>diagonalizes</u> if X is isomorphic to a direct sum of objects ((M₁,...,M_n),(f₁,...,f_n-1)) in which each M_i is of rank one. We establish an algorithm which diagonalizes any diagonalizable object X of β_n, and which fails only in case X is not diagonalizable. Let Λ be an artin algebra of finite type. We prove that for a fixed C in mod(Λ) there are only finitely many modules A in mod(Λ) (up to isomorphism) for which a short exact sequence of the form 0⟶A⟶B⟶C⟶0 is indecomposable. / Ph. D. / incomplete_metadata

The concept of unique factorization was first recognized in the 1840s, but even then, it was still fairly believed to be automatic. The error of this assumption was exposed largely through attempts to prove Pierre de Fermat's, 1601-1665, last theorem. Once mathematicians discovered that this property did not always hold, it was only natural for them to try to search for the strongest available alternative. Thus began the attempt to generalize unique factorization. Using the ascending chain condition on principle ideals, we will show the conditions under which a ring is a unique factorization domain.

This paper defines generalized binomial coefficients and shows that they can be used to generate generalized Pascal's Triangles and have properties analogous to binomial coefficients. It uses the generalized binomial coefficients to compute the Dilworth number and the Sperner number of certain rings.

Doctorat en Sciences / info:eu-repo/semantics/nonPublished

This dissertation focuses on the significance of containment relations between the above mentioned classes of ideals. The main problem considered in Chapter II is determining conditions which lead a ring to be a P-ring, D-ring, or AM-ring when every regular ideal is a P-ideal, D-ideal, or AM-ideal, respectively. We also consider containment relations between classes of regular ideals which guarantee that the ring is a quasi-valuation ring. We continue this study into the third chapter; in particular, we look at the conditions in a quasi-valuation ring which lead to a = Jr, sr - f, and a = v. Furthermore we give necessary and sufficient conditions that a ring be a discrete rank one quasi-valuation ring. For example, if R is Noetherian, then ft = J if and only if R is a discrete rank one quasi-valuation ring.

In this dissertation, we will investigate aspects of Auslander-Reiten theory adapted to the setting of systems of submodule embeddings. Using this theory, we can compute Auslander-Reiten quivers of such categories, which among other information, yields valuable information about the indecomposable objects in such a category. A main result of the dissertation is an adaptation to this situation of the Auslander and Ringel-Tachikawa Theorem which states that for an artinian ring R of finite representation type, each R-module is a direct sum of finite-length indecomposable R-modules. In cases where this applies, the indecomposable objects obtained in the Auslander-Reiten quiver give the building blocks for the objects in the category. We also briefly discuss in which cases systems of submodule embeddings form a Frobenius category, and for a few examples explore pointwise Calabi-Yau dimension of such a category. / by Audrey Moore. / Thesis (Ph.D.)--Florida Atlantic University, 2009. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2009. Mode of access: World Wide Web.

by Lok Tsan-ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 74-75). / Introduction --- p.2 / Chapter 1 --- "Rings, Modules and Categories" --- p.4 / Chapter 1.1 --- "Rings, Subrings and Ideals" --- p.5 / Chapter 1.2 --- Modules and Categories --- p.8 / Chapter 1.3 --- Module Theory --- p.13 / Chapter 2 --- Isomorphisms between Endomorphism rings of Quasiprogener- ators --- p.24 / Chapter 2.1 --- Preliminaries --- p.24 / Chapter 2.2 --- The Fundamental Theorem --- p.31 / Chapter 2.3 --- Isomorphisms Induced by Semilinear Maps --- p.41 / Chapter 2.4 --- Isomorphisms of General linear groups --- p.46 / Chapter 3 --- Endomorphism ring of projective module --- p.54 / Chapter 3.1 --- Preliminaries --- p.54 / Chapter 3.2 --- Main Theorem --- p.60 / Bibliography --- p.74