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  • Thesis

  • Authors: Diesl, Alexander James (2006)

  • A ring is called clean if every element can be written as the sum of an idempotent and a unit, and a ring is further called strongly clean if every element can be written as the sum of an idempotent and a unit that commute. The aim of this work is to study strongly clean rings by representing them as full endomorphism rings of certain modules. We are motivated by several fundamental questions that have been raised concerning strongly clean rings. We begin by placing the characterization of a strongly clean endomorphism in a more general context and then use these ideas to develop certain new classes of strongly clean rings. In Chapter 2, we investigate upper-triangular matrix rings over local rings. We show that Tn(R) is strongly clean for a large class of local rings R. We then u...

  • Thesis

  • Authors: Tomoda, Satoshi (2006)

  • In 1932, Seifert introduced a large class of 3-manifolds, now called Seifert manifolds. In his paper, he determined the fundamental groups and homology groups of Seifert manifolds. Until the year 2000, however, the ring structures of these cohomology groups were not known apart from a few special cases. Bryden, Hayat-Legrand, Zieschang, and Zvengrowski computed the cup product structures in the cohomology of orientable Seifert manifolds with infinite fundamental group and coefficients Zr (for p prime). This dissertation is primarily devoted to carrying out the above programme for those Seifert manifolds with finite fundamental group, also known as the Clifford-Klein spherical space forms, with coefficients including both Z and 74,.

  • Thesis

  • Authors: Wood, William E. (2006)

  • The main result in this thesis bounds the combinatorial modulus of a ring in a triangulation graph in terms of the modulus of a related ring. The bounds depend only on how the rings are related and not on the rings themselves. This may be used to solve the combinatorial type problem in a variety of situations, most signi cantly in graphs with unbounded degree. Other results regarding the type problem are presented along with several examples illustrating the limits of the results.

  • Thesis

  • Authors: Bartley, Katherine (2006)

  • Algebraic geometric codes over rings were defined and studied in the late 1990's by Walker, but no decoding algorithm was given. In this dissertation, we present three decoding algorithms for algebraic geometric codes over rings. The first algorithm presented is a modification of the basic algorithm for algebraic geometric codes over fields, and decodes with respect to the Hamming weight. The second algorithm presented is a modification of the Guruswami-Sudan algorithm, a list decoding algorithm for one-point algebraic geometric codes over fields. This algorithm also decodes with respect to the Hamming weight. Finally, we show how the Koetter-Vardy algorithm, a soft-decision decoding algorithm, can be used to decode one-point algebraic geometric codes over rings of the form Z/prZ, ...

  • Thesis

  • Authors: Montgomery, Martin (2006)

  • Let R be an artinian ring with identity. Denote by J = J(R) the Jacobson radical of R. The ring R is cleft if there is a subring S C R such that R = S ® J as abelian groups and S R/J as rings. Among the examples of cleft rings are finite dimensional algebras and all square-free rings. The results of Anick and Green allow for the construction of projective resolutions of simple modules over finite dimensional split algebras of the form KF/I. The resulting bound on the projective dimension of simples provides a bound on the global dimension of the algebra. In this paper, we develop the necessary ideas to apply this method to certain members of the class of binomial rings. This class, defined recently by Sklar, includes the classes of monomial rings, square-free rings, and binomial al...

  • Thesis

  • Authors: Gorton, Christine E. (2006)

  • The concept of a primary ideal in commutative rings has been extended in several ways to noncommutative ring theory. Generically these are called generalized primary conditions. We investigate the interrelations between these generalizations as well as their relation to various structural properties of a ring. Particular attention is given to the additive group structure of the ring. Conditions are given for the intersection of generalized primary ideals to be generalized primary. Ascending chain conditions on ideals are useful in this context. Various set inclusion relations and permutation identities are used to establish conditions for one-sided primary conditions to be two-sided. Examples are given to illustrate and delimit the theory developed.

  • Thesis

  • Authors: Boynton, Jason (2006)

  • Let D be an integral domain with field of fractions K, and let E be a non-empty finite subset of D. For n > 2, we show that the n-generator property forD is equivalent to the n-generator property for Int(E, D), which is equivalent to strong (n + 1)-generator property for Int(E, D). We also give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (that is, a ring whose ideals are totally ordered by inclusion), and we give necessary and sufficient conditions that the pullback of a conductor square be an arithmetical ring (that is, a ring which is locally a chain ring at every maximal ideal). We characterize all Priifer domains R between D[X] and K[X] such that the conductor C of K[X] into R is non-zero. As an application, we show that for n > 2,...

  • Thesis

  • Authors: Nielsen, Pace Peterson (2006)

  • The exchange property was introduced in 1964 by Crawley and Masson. Our work focuses on the problem they posed of when the finite exchange property implies the full exchange property. The methods of this work are two-fold. The first chapter presents a ring-theoretical construction showing that modules whose endomorphism rings are strongly 7r-regular or strongly clean have 'Zo-exchange. These methods are further refined to show that modules with Dedekind-finite, regular endomorphism rings have full exchange. The second chapter focuses on direct sum decompositions for modules. We generalize a theorem of Stock, which shows that projective modules with Zo-exchange have full exchange. We further show that one may "mod out" by the radical, when considering the exchange property for proje...

  • Thesis

  • Authors: Jiang, Yongbin (2006)

  • In algebraic number theory, the study of Galois groups is one of the subjects mathematicians are most interested in. A fruitful set of results have been obtained in this field. Class field theory and some of its consequent theorems are certainly an important part of these results. In [1], E.Artin and Tate used the notion of class formations to reinterpret class field theory and proposed another interesting object, which is called the Weil group and has a strong relation with class formations, for us to study. Recently, research shows that Weil groups deserve more investigation and they behave better than the classical Galois groups in some cases. Lichtenbaum's paper [11] showed us evidence for the above mentioned opinion. In his paper, Lichtenbaum gave the definition of Weil-kale t...