Thesis (Doctoral)--Rheinische Friedrich-Wilhelms-Universtät Bonn, 1990. / Includes bibliographical references.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 2005. / Includes vita. Includes bibliographical references (p. 90-93).

In this dissertation, we examine atomicity in rings with zero divisions. We begin by examining the relationship between a ring’s level of atomicity and the highest level of irreducibility shared by the ring’s irreducible elements. Later, we chose one of the higher forms of atomicity and identify ways of building large classes of examples of rings that rise to this level of atomicity but no higher. Characteristics of the various types of irreducible elements will also be examined. Next, we extend our view to include polynomial extensions of rings with zero divisors. In particular, we focus on properties of the three forms of maximal common divisors and how a ring’s classification as an MCD, SMCD, or VSMCD ring affects its atomicity. To conclude, we identify some unsolved problems relating to the topics discussed in this dissertation.

We define two related invariants for a d-dimensional local ring (R,m,k) called syzygy and differential symmetric signature by looking at the maximal free splitting of reflexive symmetric powers of two modules: the top dimensional syzygy module of the residue field and the module of Kähler differentials of R over k. We compute these invariants for two-dimensional ADE singularities obtaining 1/|G|, where |G| is the order of the acting group, and for cones over elliptic curves obtaining 0 for the differential symmetric signature. These values coincide with the F-signature of such rings in positive characteristic.

In this thesis we look at different topics and problems that combine the theory of combinatorics with the theory of (commutative) algebra.

Given a finite simple graph G, one can associate to G an ideal I_G, called the toric ideal of G. There are a number of algebraic invariants of ideals which are frequently studied in commutative algebra. In general, understanding these invariants is very difficult for arbitrary ideals. However, when the ideals are related to combinatorial objects, in this case, graphs, a deeper investigation can be conducted. If, in addition, the graph G is bipartite, even more can be said about these invariants. In this thesis, we explore a comparison of invariants of toric ideals of bipartite graphs. Our main result describes all possible values for the tuple (reg(K[E]/I_G), deg(h_{K[E]/I_G}), pdim(K[E]/I_G), depth(K[E]/I_G), dim(K[E]/I_G)) when G is a bipartite graph on n ≥ 1 vertices. / Thesis / Master of Science (MSc)

Groups of divisibility have played an important role in commutative algebra for many years. In 1932 Wolfgang Krull showed in [12] that every linearly ordered Abelian group can be realized as the group of divisibility of a valuation domain. Since then it has also been proven that every lattice-ordered Abelian group can be recognized as the group of divisibility of a Bezont domain. Knowing these two facts allows us to use groups of divisibility to find examples of rings with highly exotic properties. For instance, we use them here to find examples of rings which admit elements that factor uniquely as the product of uncountably many primes. In addition to allowing us to create examples, groups of divisibility can he used to characterize some of the most important rings most commonly encountered in factorization theory, including valuation domains, UFD's, GCD domains, and antimatter domains. We present some of these characterizations here in addition to using them to create many examples of our own, including examples of rings which admit chains of prime ideals in which there are uncountably many primes in the chain. Moreover, we use groups of divisibility to prove that every fragmented domain must have infinite Krull dimension.

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La théorie des pavages apériodiques a connu des développements rapides depuis les années 1980, avec la découvertes d'alliages métalliques cristallisant dans une structure quasi-périodique.Dans cette thèse, on étudie particulièrement deux méthodes de construction de pavages : par coupe et projection, et par substitution. Deux angles d'approche sont développés : l'étude de la fonction de complexité, et l'étude métrique de l'espace de pavages.Dans une première partie, on calcule l'asymptotique de la fonction de complexité pour des pavages coupe et projection, généralisant ainsi des résultats connus en dynamiques symbolique pour la dimension 1. On montre que pour un pavage coupe et projection canonique N sur d sans période, la complexité croît (à des constantes près) comme n à la puissance a, où a est un entier compris entre d et N-d.Ensuite, on se base sur une construction de Pearson et Bellissard qui construisent un triplet spectral sur les ensembles de Cantor ultramétriques. On suit leur construction dans le cas d'ensembles de Cantor auto-similaires. Elle s'applique en particulier aux transversales d'espaces de pavages de substitution.Enfin, on fait le lien entre la distance usuelle sur l'enveloppe d'un pavage et la complexité de ce pavage. Les liens entre complexité et métrique permettent de donner une preuve directe du fait suivant : la complexité des pavages de substitution apériodiques de dimension d croît comme n à la puissance d.La question de liens entre la complexité et la topologie (et pas seulement avec la distance) reste ouverte. Nous apportons cependant des réponses partielles dans cette direction. / Since the 1980s, the theory of aperiodic tilings developed quickly, motivated by the discovery of metallic alloys which crystallize in an aperiodic structure. This highlighted the need for new models of crystals.Two models of aperiodic tilings are specifically studied in this dissertation. First, the cut-and-project method, then the inflation and substitution method. Two point of view are developed for the study of these objects: the study of the complexity function associated to a tiling, and the metric study of the associated tiling space.In a first part, the asymptotic behaviour of the complexity function for cut-and-project tilings is studied. The results stated here generalize formerly known results in the specific case of dimension 1. It is proved that for an (N,d) canonical projection tiling without periods, the complexity grows like n to the a, with a an integer greater or equal to d but lesser or equal to N-d.A second part is based on a construction by Pearson and Bellissard of a spectral triple for ultrametric Cantor sets. Their construction is applied to self-similar Cantor sets. It applies in particular to the transversal of substitution tiling spaces.In a last part, the links between the complexity function of a tiling and the usual distance on its associated tiling space are made explicit. These links can provide a direct and complete proof of the following fact: the complexity of an aperiodic d-dimensional substitution tiling grows asymptotically as n to the d, up to constants. These links between complexity and distance raises the question of links between complexity and topology. Partial answers are given in this direction.

No description available.