This thesis is a study of some properties of prime ideals in commutative rings with unity.

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This thesis investigates some of the properties of R-modules. The material is presented in three chapters. Definitions and theorems which are assumed are stated in Chapter I. Proofs of these theorems may be found in Zariski and Samuel, Commutative Algebra, Vol. I, 1958. It is assumed that the reader is familiar with the basic properties of commutative rings and ideals in rings. Properties of R-modules are developed in Chapter II. The most important results presented in this chapter include existence theorems for R-modules and properties of submodules in R-modules. The third and final chapter presents an example which illustrates how a ring R, may be regarded as an R-module and speaks of the direct sum of ideals of a ring as a direct sum of submodules.

This dissertation focuses on the properties of a domain which has the property that each ideal is a finite intersection of a π-ideal, the properties of a domain which have the property that each ideal is a finite product of π-ideal, and the containment relations of the resulting classes of ideals. Chapter 1 states definitions which are needed in later chapters. Chapters 2 and 3 focuses on domains which have the property that each ideal in D is a finite intersection of π-ideals while Chapter 4 focuses on domains with the property that each ideal is a finite product of π-ideals. Chapter 5 discusses the containment relations which occur as a result of Chapters 2 and 3.

In [2], Dan Anderson and Andrea Frazier introduced a generalized theory of factorization. Given a relation τ on the nonzero, nonunit elements of an integral domain D, they defined a τ-factorization of a to be any proper factorization a = λa1 · · · an where λ is in U (D) and ai is τ-related to aj, denoted ai τ aj, for i not equal to j . From here they developed an abstract theory of factorization that generalized factorization in the usual sense. They were able to develop a number of results analogous to results already known for usual factorization. Our work focuses on the notion of τ-factorization when the relation τ has characteristics similar to those of coprimeness. We seek to characterize such τ-factorizations. For example, let D be an integral domain with nonzero, nonunit elements a, b ∈ D. We say that a and b are comaximal (resp. v-coprime, coprime ) if (a, b) = D (resp., (a, b)v = D, [a, b] = 1). More generally, if ∗ is a star-operation on D, a and b are ∗-coprime if (a, b)∗ = D. We then write a τmax b (resp. a τv b, a τ[ ] b, or a τ∗ b) if a and b are comaximal (resp. v -coprime, coprime, or ∗-coprime).

In this thesis, we prove unconditional lower bounds on resources needed to compute explicit functions in the following three models of computation: constant-depth boolean circuits, multivariate polynomials over commutative rings and the 'Number on the Forehead' model of multiparty communication. Apart from using tools from diverse areas, we exploit the rich interplay between these models to make progress on questions arising in the study of each of them. / Boolean circuits are natural computing devices and are ubiquitous in the modern electronic age. We study the limitation of this model when the depth of circuits is fixed, independent of the length of the input. The power of such constant-depth circuits using gates computing modular counting functions remains undetermined, despite intensive efforts for nearly twenty years. We make progress on two fronts: let m be a number having r distinct prime factors none of which divides ℓ. We first show that constant depth circuits employing AND/OR/MODm gates cannot compute efficiently the MAJORITY and MODℓ function on n bits if 'few' MODm gates are allowed, i.e. they need size nW&parl0;1s&parl0;log n&parr0;1/&parl0;r-1&parr0;&parr0; if s MODm gates are allowed in the circuit. Second, we analyze circuits that comprise only MOD m gates, We show that in sub-linear size (and arbitrary depth), they cannot compute AND of n bits. Further, we establish that in that size they can only very poorly approximate MODℓ. / Our first result on circuits is derived by introducing a novel notion of computation of boolean functions by polynomials. The study of degree as a resource in polynomial representation of boolean functions is of much independent interest. Our notion, called the weak generalized representation, generalizes all previously studied notions of computation by polynomials over finite commutative rings. We prove that over the ring Zm , polynomials need Wlogn 1/r-1 degree to represent, in our sense, simple functions like MAJORITY and MODℓ. Using ideas from arguments in communication complexity, we simplify and strengthen the breakthrough work of Bourgain showing that functions computed by o(log n)-degree polynomials over Zm do not even correlate well with MODℓ. / Finally, we study the 'Number on the Forehead' model of multiparty communication that was introduced by Chandra, Furst and Lipton [CFL83]. We obtain fresh insight into this model by studying the class CCk of languages that have constant k-party deterministic communication complexity under every possible partition of input bits among parties. This study is motivated by Szegedy's [Sze93] surprising result that languages in CC2 can all be extremely efficiently recognized by very shallow boolean circuits. In contrast, we show that even CC 3 contains languages of arbitrarily large circuit complexity. On the other hand, we show that the advantage of multiple players over two players is significantly curtailed for computing two simple classes of languages: languages that have a neutral letter and those that are symmetric. / Extending the recent breakthrough works of Sherstov [She07, She08b] for two-party communication, we prove strong lower bounds on multiparty communication complexity of functions. First, we obtain a bound of n O(1) on the k-party randomized communication complexity of a function that is computable by constant-depth circuits using AND/OR gates, when k is a constant. The bound holds as long as protocols are required to have better than inverse exponential (i.e. 2-no1 ) advantage over random guessing. This is strong enough to yield lower bounds on the size of an important class of depth-three circuits: circuits having a MAJORITY gate at its output, a middle layer of gates computing arbitrary symmetric functions and a base layer of arbitrary gates of restricted fan-in. / Second, we obtain nO(1) lower bounds on the k-party randomized (bounded error) communication complexity of the Disjointness function. This resolves a major open question in multiparty communication complexity with applications to proof complexity. Our techniques in obtaining the last two bounds, exploit connections between representation by polynomials over teals of a boolean function and communication complexity of a closely related function.

For any ring R (associative with 1) we associate a space X of prime torsion theories endowed with Golan's SBO-topology. A separated presheaf L(-,M) on X is then constructed for any right R-module M$ sb{ rm R}$, and a sufficient condition on M is given such that L(-,M) is actually a sheaf. The sheaf space rm E { buildrel{ rm p} over longrightarrow} X) etermined by L(-,M) represents M in the following sense: M is isomorphic to the module of continuous global sections of p. These results are applied to the right R-module R$ sb{ rm R}$ and it is seen that semiprime rings satisfy the required condition for L(-,R) to be a sheaf. Among semiprime rings two classes are singled out, fully symmetric semiprime and right noetherian semiprime rings; these two kinds of rings have the desirable property of yielding "nice" stalks for the above sheaf.

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Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 106-108).