We say that a commutative ring R has the unique decomposition into ideals (UDI) property if, for any R-module which decomposes into a _nite direct sum of indecomposable ideals, this decomposition is unique up to the order and isomorphism class of the ideals. In a 2001 paper, Goeters and Olberding characterize the UDI property for Noetherian integral domains and in a 2011 paper Ay and Klingler obtain similar results for Noetherian reduced rings. We characterize the UDI property for Noetherian rings in general. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2008. / Title from title screen (site viewed Oct. 31, 2008). PDF text: v, 54 p. : ill. ; 769 K. UMI publication number: AAT 3313102. Includes bibliographical references. Also available in microfilm and microfiche formats.

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.

Orientador: Reginaldo Palazzo Junior / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-09T21:06:53Z (GMT). No. of bitstreams: 1 Sampaio_IngridAraujo_M.pdf: 836393 bytes, checksum: c88f5bde14a891b8579e6d9dca463a95 (MD5) Previous issue date: 2007 / Resumo: Neste trabalho apresentamos algumas relações existentes entre codigos c'clicos e a transformada discreta de Fourier ambos sobre aneis locais. Para isso, 'e necessario a identificação do grupo das unidades associado a cada um dos anéis considerados. Como consequencia, codigos ciclicos sobre tais aneis podem ser construidos. Em seguida, construimos geradores de sequencias atravees dos registros de deslocamento com realimentação linear (LFSR), a partir dos polinomios geradores, cujos coeficientes pertencem a um corpo finito e a um anel comutativo finito local com identidade. Finalmente, realizamos a transformada discreta de Fourier por meio do polinomio gerador dos codigos ciclicos sobre aneis locais / Abstract: In this research we present some existing relationships between cyclic codes and discrete Fourier transform both local rings. For this, it is necessary to identify the groups of unit associated with each corresponding local ring. As a consequence, cyclic codes over these rings may be constructed. Next, we construct sequence generators by use of linear feedback shift register (LFSR), from generator polynomials whose coefficients belong either to finite field or to a local finite commutative ring with identity. Finally, the discrete Fourier transform is realized by use of the generator polynomial of cyclic codes over local rings / Mestrado / Telecomunicações e Telemática / Mestre em Engenharia Elétrica

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Dans cette thèse, nous nous intéressons à l'arithmétique de certains corps de fonctions. Nous cherchons à établir dans un premier temps des théorèmes de dualité arithmétique sur ces corps, pour les appliquer ensuite à l'étude des points rationnels sur certaines variétés algébriques. Dans les trois premiers chapitres, nous travaillons sur le corps des fonctions d'une courbe sur un corps local supérieur (comme Qp, Qp((t)), C((t)) ou C((t))((u))). Dans le premier chapitre, nous établissons sur un tel corps des théorèmes de dualité arithmétique « à la Poitou-Tate » pour les modules finis, les tores, et même pour certains complexes de tores. Nous montrons aussi l'existence, sous certaines hypothèses, de certaines portions des suites exactes de Poitou-Tate correspondantes. Ces résultats sont appliqués dans le deuxième chapitre à l'étude du principe local-global pour les algèbres simples centrales, de l'approximation faible pour les tores, et des obstructions au principe local-global pour les torseurs sous des groupes linéaires connexes. Dans le troisième chapitre, nous nous penchons sur les variétés abéliennes et établissons des théorèmes de dualité arithmétique « à la Cassels-Tate ». Cela demande aussi de mener une étude fine des variétés abéliennes sur les corps locaux supérieurs. Dans le quatrième et dernier chapitre, nous travaillons sur les corps des fractions de certaines algèbres locales normales de dimension 2 (typiquement C((x, y)) ou Fp((x, y))). Nous établissons d'abord un théorème de dualité en cohomologie étale « à la Artin-Verdier » dans ce contexte. Cela nous permet ensuite de montrer des théorèmes de dualité arithmétique en cohomologie galoisienne « à la Poitou-Tate » pour les modules finis et les tores. Nous appliquons finalement ces résultats à l'étude de l'approximation faible pour les tores et des obstructions au principe local-global pour les torseurs sous des groupes linéaires connexes. / In this thesis, we are interested in the arithmetic of some function fields. We first want to establish arithmetic duality theorems over those fields, in order to apply them afterwards to the study of rational points on algebraic varieties. In the first three chapters, we work on the function field of a curve defined over a higher-dimensional local field (such as Qp, Qp((t)), C((t)) or C((t))((u))). In the first chapter, we establish "Poitou-Tate type" arithmetic duality theorems over such fields for finite modules, tori and even some complexes of tori. We also prove the existence, under some hypothesis, of parts of the corresponding Poitou-Tate exact sequences. These results are applied in the second chapter to the study of the local-global principle for central simple algebras, of weak approximation for tori, and of obstructions to local-global principle for torsors under connected linear algebraic groups. In the third chapter, we are interested in abelian varieties and we establish "Cassels-Tate type" arithmetic duality theorems. To do so, we also need to carry out a precise study of abelian varieties over higher-dimensional local fields. In the fourth and last chapter, we work on the field of fractions of some 2-dimensional normal local algebras (such as C((x, y)) or Fp((x, y))). We first establish in this context an "Artin-Verdier type" duality theorem in étale cohomology. This allows us to prove "Poitou-Tate type" arithmetic duality theorems in Galois cohomology for finite modules and tori. In the end, we apply these results to the study of weak approximation for tori and of obstructions to local-global principle for torsors under connected linear algebraic groups.