To see the other types of publications on this topic, follow the link: Commutative rings.
Dissertations / Theses on the topic 'Commutative rings'
Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles
Consult the top 50 dissertations / theses for your research on the topic 'Commutative rings.'
Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.
You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.
Browse dissertations / theses on a wide variety of disciplines and organise your bibliography correctly.
1
Margolin, Benjamin Paul. "Non-commutative deformation rings." Diss., University of Iowa, 2016. https://ir.uiowa.edu/etd/3135.
Full textAbstract:
The goal of this thesis is to study non-commutative deformation rings of representations of algebras. The main motivation is to provide a generalization of the deformation theory over commutative local rings studied by B. Mazur, M. Schlessinger and others. The latter deformation theory has played an important role in number theory, and in particular in the proof of Fermat's Last Theorem.
The thesis is divided into two parts.
In the first part, A is an arbitrary λ-algebra for a complete local commutative Noetherian ring λ with residue field k. A category Ĉ is defined whose objects are complete local λ-algebras R with residue field k such that R is a quotient ring of a power series algebra over λ in finitely many non-commuting variables. If V is a finite dimensional k-vector space that is also a left A-module and that satisfies a natural finiteness condition, it is proved that V has a so-called versal deformation ring R(A,V). More precisely, R(A,V) is an object in Ĉ such that the isomorphism class of every lift of V over an object R in Ĉ arises from a morphism α : R(A,V)→ R in Ĉ and α is unique if R is the ring of dual numbers k[ϵ].
In the second part, two particular examples of λ, A and V are studied and the versal deformation ring R(A,V) is determined in each of these cases. In the first example, λ=k, A is a series of non-commutative k-algebras depending on a parameter r≥2, and V is a particular quotient module of A; it is shown that R(A,V) is isomorphic to A. The second example builds on the first example when r=2 and uses that, if additionally the characteristic of k is 2, then A is isomorphic to the group ring k[D8] of a dihedral group D8 of order 8.
It is shown that if k is perfect and W is the ring of infinite Witt vectors over k, then R(W[D8],V) is isomorphic to W[D8].
2
Sekaran, Rajakrishnar. "Fuzzy ideals in commutative rings." Thesis, Rhodes University, 1995. http://hdl.handle.net/10962/d1005221.
Full textAbstract:
In this thesis, we are concerned with various aspects of fuzzy ideals of commutative rings. The central theorem is that of primary decomposition of a fuzzy ideal as an intersection of fuzzy primary ideals in a commutative Noetherian ring. We establish the existence and the two uniqueness theorems of primary decomposition of any fuzzy ideal with membership value 1 at the zero element. In proving this central result, we build up the necessary tools such as fuzzy primary ideals and the related concept of fuzzy maximal ideals, fuzzy prime ideals and fuzzy radicals. Another approach explores various characterizations of fuzzy ideals, namely, generation and level cuts of fuzzy ideals, relation between fuzzy ideals, congruences and quotient fuzzy rings. We also tie up several authors' seemingly different definitions of fuzzy prime, primary, semiprimary and fuzzy radicals available in the literature and show some of their equivalences and implications, providing counter-examples where certain implications fail.
3
Hasse, Erik Gregory. "Lowest terms in commutative rings." Diss., University of Iowa, 2018. https://ir.uiowa.edu/etd/6433.
Full textAbstract:
Putting fractions in lowest terms is a common problem for basic algebra courses, but it is rarely discussed in abstract algebra. In a 1990 paper, D.D. Anderson, D.F. Anderson, and M. Zafrullah published a paper called Factorization in Integral Domains, which summarized the results concerning different factorization properties in domains. In it, they defined an LT domain as one where every fraction is equal to a fraction in lowest terms. That is, for any x/y in the field of fractions of D, there is some a/b with x/y=a/b and the greatest common divisor of a and b is 1. In addition, R. Gilmer included a brief exercise concerning lowest terms over a domain in his book Multiplicative Ideal Theory.
In this thesis, we expand upon those definitions. First, in Chapter 2 we make a distinction between putting a fraction in lowest terms and reducing it to lowest terms. In the first case, we simply require the existence of an equal fraction which is in lowest terms, while the second requires an element which divides both the numerator and the denominator to reach lowest terms. We also define essentially unique lowest terms, which requires a fraction to have only one lowest terms representation up to unit multiples. We prove that a reduced lowest terms domain is equivalent to a weak GCD domain, and that a domain which is both a reduced lowest terms domain and a unique lowest terms domain is equivalent to a GCD domain. We also provide an example showing that not every domain is a lowest terms domain as well as an example showing that putting a fraction in lowest terms is a strictly weaker condition than reducing it to lowest terms.
Next, in Chapter 3 we discuss how lowest terms in a domain interacts with the polynomial ring. We prove that if D[T] is a unique lowest terms domain, then D must be a GCD domain. We also provide an alternative approach to some of the earlier results using the group of divisibility.
So far, all fractions have been representatives of the field of fractions of a domain. However, in Chapter 4 we examine fractions in other localizations of a domain. We define a necessary and sufficient condition on the multiplicatively closed set, and then examine how this relates to existing properties of multiplicatively closed sets.
Finally, in Chapter 5 we briefly examine lowest terms in rings with zero divisors. Because many properties of GCDs do not hold in such rings, this proved difficult. However, we were able to prove some results from Chapter 2 in this more general case.
4
Vo, Monika. "New classes of finite commutative rings." Thesis, University of Hawaii at Manoa, 2003. http://proquest.umi.com/pqdweb?index=2&did=765961151&SrchMode=1&sid=1&Fmt=2&VInst=PROD&VType=PQD&RQT=309&VName=PQD&TS=1208558919&clientId=23440.
Full text
5
Akalan, Evrim. "Generalizations of non-commutative uniquefactorization rings." Thesis, University of Warwick, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.502119.
Full textAbstract:
There is a well-developed theory of unique factorization domains in commutative algebra. The generalization of this concept to non-commutative rings has also been extensively studied (e.g. in [19], [16], [1]). This thesis is concerned with classes of non-commutative rings which are generalizations of non-commutative Noetherian unique factorization rings. Noetherian UFR's are maximal orders and every reflexive ideal is invertible in these rings. Clearly maximal orders and reflexive ideals are important concepts and we examine them in this thesis. UFR's can also be considered as rings in which the divisor class group is trivial. This provides the motivation for us to study this group more generally. In Chapter 1, we give the basic material we shall need from the theory of noncommutative rings-and in Chapter 2, we present the known results about certain classes of rings which are crucial for this thesis. Chapters 3, 4 and 5 contain the original work of this thesis. In Chapter 3, we study the prime Noetherian maximal orders with enough invertible ideals. We show that in such rings every height 1 prime ideal is maximal reflexive and we prove results which generalize some of the results of Asano orders. In Chapter 4, we investigate divisor class groups. We also study the relations between these groups and the divisor -class group of the centre of the ring. In Chapter 5, we introduce the Generalized Dedekind prime rings (G-Dedekind prime rings), which are also a generalization of Noetherian UFR's, and we study this class of rings.
6
Rumbos, Irma Beatriz. "A sheaf representation for non-commutative rings /." Thesis, McGill University, 1987. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=70356.
Full textAbstract:
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.
7
Denizler, Ismail Hakki. "Artinian modules over commutative rings, and duality." Thesis, University of Sheffield, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.364185.
Full text
8
Granger, Ginger Thibodeaux. "Properties of R-Modules." Thesis, University of North Texas, 1989. https://digital.library.unt.edu/ark:/67531/metadc500710/.
Full textAbstract:
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.
9
Mooney, Christopher Park. "Generalized factorization in commutative rings with zero-divisors." Thesis, The University of Iowa, 2013. http://pqdtopen.proquest.com/#viewpdf?dispub=3595128.
Full textAbstract:
The study of factorization in integral domains has a long history. Unique factorization domains, like the integers, have been studied extensively for many years. More recently, mathematicians have turned their attention to generalizations of this such as Dedekind domains or other domains which have weaker factorization properties. Many authors have sought to generalize the notion of factorization in domains. One particular method which has encapsulated many of the generalizations into a single study is that of τ-factorization, studied extensively by A. Frazier and D.D. Anderson.
Another generalization comes in the form of studying factorization in rings with zero-divisors. Factorization gets quite complicated when zero-divisors are present due to the existence of several types of associate relations as well as several choices about what to consider the irreducible elements.
In this thesis, we investigate several methods for extending the theory of τ-factorization into rings with zero-divisors. We investigate several methods including: 1) the approach used by A.G. Agˇargün and D.D. Anderson, S. Chun and S. Valdes-Leon in several papers; 2) the method of U-factorization developed by C.R. Fletcher and extended by M. Axtell, J. Stickles, and N. Baeth and 3) the method of regular factorizations and 4) the method of complete factorizations.
This thesis synthesizes the work done in the theory of generalized factorization and factorization in rings with zero-divisors. Along the way, we encounter several nice applications of the factorization theory. Using τ z-factorizations, we discover a nice relationship with zero-divisor graphs studied by I. Beck as well as D.D. Anderson, D.F. Anderson, A. Frazier, A. Lauve, and P. Livingston. Using τ-U-factorization, we are able to answer many questions that arise when discussing direct products of rings.
There are several benefits to the regular factorization factorization approach due to the various notions of associate and irreducible coinciding on regular elements greatly simplifying many of the finite factorization property relationships. Complete factorization is a very natural and effective approach taken to studying factorization in rings with zero-divisors. There are several nice results stemming from extending τ-factorization in this way. Lastly, an appendix is provided in which several examples of rings satisfying the various finite factorization properties studied throughout the thesis are given.
10
Coutinho, S. C. "Generating modules efficiently over non-commutative noetherian rings." Thesis, University of Leeds, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.372589.
Full text
11
Hayes, Leslie Danielle. "The plus closure of an ideal." Access restricted to users with UT Austin EID Full text (PDF) from UMI/Dissertation Abstracts International, 2001. http://wwwlib.umi.com/cr/utexas/fullcit?p3035948.
Full text
12
Emerson, Sharon Sue. "Overrings of an Integral Domain." Thesis, University of North Texas, 1992. https://digital.library.unt.edu/ark:/67531/metadc332679/.
Full textAbstract:
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.
13
Volkweis, Leite Samuel [Verfasser]. "p-adic Representations of Commutative Rings / Samuel Volkweis Leite." Konstanz : Bibliothek der Universität Konstanz, 2013. http://d-nb.info/1031879943/34.
Full text
14
Bell, Kathleen. "Cayley Graphs of PSL(2) over Finite Commutative Rings." TopSCHOLAR®, 2018. https://digitalcommons.wku.edu/theses/2102.
Full textAbstract:
Hadwiger's conjecture is one of the deepest open questions in graph theory, and Cayley graphs are an applicable and useful subtopic of algebra.
Chapter 1 will introduce Hadwiger's conjecture and Cayley graphs, providing a summary of background information on those topics, and continuing by introducing our problem. Chapter 2 will provide necessary definitions. Chapter 3 will give a brief survey of background information and of the existing literature on Hadwiger's conjecture, Hamiltonicity, and the isoperimetric number; in this chapter we will explore what cases are already shown and what the most recent results are. Chapter 4 will give our decomposition theorem about PSL2 (R). Chapter 5 will continue with corollaries of the decomposition theorem, including showing that Hadwiger's conjecture holds for our Cayley graphs. Chapter 6 will finish with some interesting examples.
15
Chattopadhyay, Arkadev. "Circuits, communication and polynomials." Thesis, McGill University, 2008. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=115660.
Full textAbstract:
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.
16
Edmonds, Ranthony A. C. "Factorization in polynomial rings with zero divisors." Diss., University of Iowa, 2018. https://ir.uiowa.edu/etd/3248.
Full textAbstract:
Factorization theory is concerned with the decomposition of mathematical objects. Such an object could be a polynomial, a number in the set of integers, or more generally an element in a ring. A classic example of a ring is the set of integers. If we take any two integers, for example 2 and 3, we know that $2 \cdot 3=3\cdot 2$, which shows that multiplication is commutative. Thus, the integers are a commutative ring. Also, if we take any two integers, call them $a$ and $b$, and their product $a\cdot b=0$, we know that $a$ or $b$ must be $0$. Any ring that possesses this property is called an integral domain. If there exist two nonzero elements, however, whose product is zero we call such elements zero divisors. This thesis focuses on factorization in commutative rings with zero divisors.
In this work we extend the theory of factorization in commutative rings to polynomial rings with zero divisors. For a commutative ring $R$ with identity and its polynomial extension $R[X]$ the following questions are considered: if one of these rings has a certain factorization property, does the other? If not, what conditions must be in place for the answer to be yes? If there are no suitable conditions, are there counterexamples that demonstrate a polynomial ring can possess one factorization property and not another? Examples are given with respect to the properties of atomicity and ACCP. The central result is a comprehensive characterization of when $R[X]$ is a unique factorization ring.
17
Oman, Gregory Grant. "A generalization of Jónsson modules over commutative rings with identity." Columbus, Ohio : Ohio State University, 2006. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1164331653.
Full text
18
Bradford, Jeremy. "Commutative endomorphism rings of simple abelian varieties over finite fields." Thesis, University of Maryland, College Park, 2013. http://pqdtopen.proquest.com/#viewpdf?dispub=3557641.
Full textAbstract:
In this thesis we look at simple abelian varieties defined over a finite field k = [special characters omitted]pn with Endk( A) commutative. We derive a formula that connects the p -rank r(A) with the splitting behavior of p in E = [special characters omitted](π), where π is a root of the characteristic polynomial of the Frobenius endomorphism. We show how this formula can be used to explicitly list all possible splitting behaviors of p in [special characters omitted]E, and we do so for abelian varieties of dimension less than or equal to four defined over [special characters omitted]p. We then look for when p divides [[special characters omitted]E : [special characters omitted][π, π]]. This allows us to prove that the endomorphism ring of an absolutely simple abelian surface is maximal at p when p ≥ 3. We also derive a condition that guarantees that p divides [[special characters omitted]E: [special characters omitted][π, π]]. Last, we explicitly describe the structure of some intermediate subrings of p-power index between [special characters omitted][π, π] and [special characters omitted]E when A is an abelian 3-fold with r(A) = 1.
19
Oman, Gregory Grant. "A generalization of Jónsson modules over commutative rings with identity." The Ohio State University, 2006. http://rave.ohiolink.edu/etdc/view?acc_num=osu1164331653.
Full text
20
Coughlin, Heather. "Classes of normal monomial ideals /." view abstract or download file of text, 2004. http://wwwlib.umi.com/cr/uoregon/fullcit?p3147816//.
Full textAbstract:
Thesis (Ph. D.)--University of Oregon, 2004.Typescript. Includes vita and abstract. Includes bibliographical references (leaves 85-86). Also available for download via the World Wide Web; free to University of Oregon users.
21
Trentham, Stacy Michelle. "Atomicity in Rings with Zero Divisors." Diss., North Dakota State University, 2011. https://hdl.handle.net/10365/28905.
Full textAbstract:
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.
22
Trentham, William Travis. "Applications of Groups of Divisibility and a Generalization of Krull Dimension." Diss., North Dakota State University, 2011. https://hdl.handle.net/10365/28904.
Full textAbstract:
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.
23
Green, Ellen Yvonne. "Characterizing the strong two-generators of certain Noetherian domains." CSUSB ScholarWorks, 1997. https://scholarworks.lib.csusb.edu/etd-project/1539.
Full text
24
Reinkoester, Jeremiah N. "Relative primeness." Diss., University of Iowa, 2010. https://ir.uiowa.edu/etd/585.
Full textAbstract:
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).
25
Race, Denise T. (Denise Tatsch). "Containment Relations Between Classes of Regular Ideals in a Ring with Few Zero Divisors." Thesis, North Texas State University, 1987. https://digital.library.unt.edu/ark:/67531/metadc331394/.
Full textAbstract:
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.
26
Petin, Burkhard. "Ein kombinatorisches Beweisverfahren für produktrelationen zwischen Gauss-summen über endlichen kommutativen Ringen." Bonn : [s.n.], 1990. http://catalog.hathitrust.org/api/volumes/oclc/24807249.html.
Full text
27
Malec, Sara. "Intersection Algebras and Pointed Rational Cones." Digital Archive @ GSU, 2013. http://digitalarchive.gsu.edu/math_diss/14.
Full textAbstract:
In this dissertation we study the algebraic properties of the intersection algebra of two ideals I and J in a Noetherian ring R. A major part of the dissertation is devoted to the finite generation of these algebras and developing methods of obtaining their generators when the algebra is finitely generated.
We prove that the intersection algebra is a finitely generated R-algebra when R is a Unique Factorization Domain and the two ideals are principal, and use fans of cones to find the algebra generators. This is done in Chapter 2, which concludes with introducing a new class of algebras called fan algebras.
Chapter 3 deals with the intersection algebra of principal monomial ideals in a polynomial ring, where the theory of semigroup rings and toric ideals can be used. A detailed investigation of the intersection algebra of the polynomial ring in one variable is obtained. The intersection algebra in this case is connected to semigroup rings associated to systems of linear diophantine equations with integer coefficients, introduced by Stanley.
In Chapter 4, we present a method for obtaining the generators of the intersection algebra for arbitrary monomial ideals in the polynomial ring.
28
Salt, Brittney M. "MONOID RINGS AND STRONGLY TWO-GENERATED IDEALS." CSUSB ScholarWorks, 2014. https://scholarworks.lib.csusb.edu/etd/31.
Full textAbstract:
This paper determines whether monoid rings with the two-generator property have the strong two-generator property. Dedekind domains have both the two-generator and strong two-generator properties. How common is this? Two cases are considered here: the zero-dimensional case and the one-dimensional case for monoid rings. Each case is looked at to determine if monoid rings that are not PIRs but are two-generated have the strong two-generator property. Full results are given in the zero-dimensional case, however only partial results have been found for the one-dimensional case.
29
Zagrodny, Christopher Michael. "Algebraic Concepts in the Study of Graphs and Simplicial Complexes." Digital Archive @ GSU, 2006. http://digitalarchive.gsu.edu/math_theses/7.
Full textAbstract:
This paper presents a survey of concepts in commutative algebra that have applications to topology and graph theory. The primary algebraic focus will be on Stanley-Reisner rings, classes of polynomial rings that can describe simplicial complexes. Stanley-Reisner rings are defined via square-free monomial ideals. The paper will present many aspects of the theory of these ideals and discuss how they relate to important constructions in commutative algebra, such as finite generation of ideals, graded rings and modules, localization and associated primes, primary decomposition of ideals and Hilbert series. In particular, the primary decomposition and Hilbert series for certain types of monomial ideals will be analyzed through explicit examples of simplicial complexes and graphs.
30
Byun, Eui Won James. "Affine varieties, Groebner basis, and applications." CSUSB ScholarWorks, 2000. https://scholarworks.lib.csusb.edu/etd-project/1611.
Full text
31
Le, Gros Giovanna. "Minimal approximations for cotorsion pairs generated by modules of projective dimension at most one over commutative rings." Doctoral thesis, Università degli studi di Padova, 2019. http://hdl.handle.net/11577/3423180.
Full textAbstract:
In this thesis we study cotorsion pairs (A, B) generated by classes of R-modules of projective dimension at most one. We are interested in when these cotorsion pairs provide covers or envelopes over commutative rings. More precisely, we investigate Enochs' Conjecture in this setting. That is, for a class A contained in the class of modules of projective dimension at most one, denoted P_1, we investigate the question of whether A is covering necessarily implies that A is closed under direct limits. Additionally, under certain restrictions we characterise the rings which satisfy this property. To this end, there were two cases to consider: when the cotorsion pair is of finite type and when it is not of finite type.
For the case that the cotorsion pair (P_1, B) is not (necessarily) of finite type, we show that over a semihereditary ring R, if P_1 is covering it must be closed under direct limits. This gives an example of a cotorsion pair not of finite type which satisfies Enochs' Conjecture.
The next part of the thesis is dedicated toward cotorsion pairs of finite type, specifically the 1-tilting cotorsion pairs over commutative rings. We rely heavily on work of Hrbek who characterises these cotorsion pairs over commutative rings, as well as work of Positselski and Bazzoni-Positselski in their work on contramodules.
We consider the case of a 1-tilting cotorsion pair (A, T) over a commutative ring with an associated Gabriel topology G, and begin by investigating when T is an enveloping class. We find that if T is enveloping, then the associated Gabriel topology must arise from a perfect localisation. That is, G must arise from a flat ring epimorphism from R to R_G, where R_G is the ring of quotients of R with respect to G. Furthermore, if G arises from a perfect localisation, T is enveloping in Mod-R if and only if the projective dimension of R_G is less than or equal to one and R/J is a perfect ring for every ideal J in G if and only if the projective dimension of R_G is less than or equal to one and the topological ring End(R_G/R) is pro-perfect.
Next, we consider the case that A is a covering class, and we prove that A is covering in Mod-R if and only if the projective dimension of R_G is less than or equal to one and both the localisation R_G is a perfect ring and R/J is a perfect ring for every J in G.
Additionally, we study general cotorsion pairs, as well as conditions for an approximation to be a minimal approximation. Moreover, we consider a hereditary cotorsion pair and show that if it provides covers it must provide envelopes.In questa tesi studiamo le coppie di cotorsione (A, B) generate da classi di R-moduli di dimensione proiettiva al più uno. Siamo interessati nel caso in cui queste coppie di cotorsione ammettano ricoprimenti o inviluppi su anelli commutativi. Più precisamente, indaghiamo la congettura di Enochs per A. Cioè, per A contenuta nella classe P_1, che denota la classe di R-moduli di dimensione proiettiva al più uno, cerchiamo di capire se per A una classe ricoprente allora necessariamente implica che A è chiusa per limiti diretti. In più, con certe restrizioni, descriviamo gli anelli che soddisfano questa proprietà. Ci sono due casi da considerare: il caso di coppia di cotorsione di tipo finito e il caso non di tipo finito. Quando la coppia di cotorsione non è (necessariamente) di tipo finito, dimostriamo che per un anello commutativo semiereditario R, se P_1 è una classe ricoprente, deve essere chiusa per limiti diretti. Questo ci da un esempio di una coppia di cotorsione che non è di tipo finito che soddisfa la congettura di Enochs. Successivamente, analizziamo le coppie di cotorsione di tipo finito. Specificamente, le coppie di cotorsione 1-tilting su anelli commutativi. A questo scopo sono indispensabili il lavoro di Hrbek, che caratterizza tali coppie di cotorsione su anelli commutativi, e il lavoro di Positselski e Bazzoni-Positselski nel loro lavoro sui contramoduli. Consideriamo il caso di una coppia di cotorsione 1-tilting (A, T) su un anello commutativo con una topologia di Gabriel associata G, e studiamo quando (A, T) ammette inviluppi. Troviamo che se T ammette inviluppi, G è una topologia di Gabriel perfetta. Cioè, G viene da un epimorfismo piatto di anelli da R a R_G dove R_G è la localizzazione di R rispetto a G. Inoltre, se G è una topologia di Gabriel perfetta, T ammette inviluppi se e solo se R_G ha dimensione proiettiva al più uno e R/J è un anello perfetto per tutti gli ideali J in G se e solo se R_G ha dimensione proiettiva al più uno e l'anello topologico End(R_G/R) è pro-perfetto. Poi consideriamo il caso in cui A è ricoprente. Dimostriamo che A è ricoprente in Mod-R se e solo se R_G ha dimensione proiettiva al più uno e R_G è un anello perfetto e R/J è perfetto per ogni J in G. In aggiunta, studiamo coppie di cotorsione in generale e studiamo condizioni sufficienti affinchè una approssimazione sia minimale. Inoltre, consideriamo una coppia di cotorsione ereditaria e dimostriamo che se ammette ricoprimenti deve ammettere inviluppi.
32
Oyinsan, Sola. "Primary decomposition of ideals in a ring." CSUSB ScholarWorks, 2007. https://scholarworks.lib.csusb.edu/etd-project/3289.
Full textAbstract:
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.
33
Diaz, Noguera Maribel del Carmen. "Sobre derivações localmente nilpotentes dos aneis K[x,y,z] e K[x,y]." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306307.
Full textAbstract:
Orientador: Paulo Roberto BrumattiDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Ciencia da Computação
Made available in DSpace on 2018-08-09T23:37:45Z (GMT). No. of bitstreams: 1 DiazNoguera_MaribeldelCarmen_M.pdf: 632573 bytes, checksum: fbcf2bd0092558fce4ba4d082d4c68c7 (MD5) Previous issue date: 2007
Resumo: O principal objetivo desta dissertação é apresentar resultados centrais sobre derivações localmente nilpotentes no anel de polinômios B = k[x1, ..., xn], para n = 3 que foram apresentados por Daniel Daigle em [2 ], [3] e [4] .Para este propósito, introduziremos os conceitos básicos e fundamentais da teoria das derivações num anel e apresentaremos resultados em relação a derivações localmente nilpotentes num domínio de característica zero e de fatorização única. Entre tais resultados está a fórmula Jacobiana que usaremos para descrever o conjunto das derivações equivalentes e localmente nilpotentes de B = k[x, y, z] e o conjunto LND(B), com B = k[x,y]. Também, explicítam-se condições equivalentes para a existência de uma derivação ?-homogênea e localmente nilpotente de B = k[x, y, z] com núcleo k[¿, g], onde {¿}, {g} e B, mdc(?) = mdc(?(¿), ? (g)) = 1
Abstract: In this dissertation we present centraIs results on locally nilpotents derivations in a ring of polynomials B = k[x1, ..., xn], for n = 3, which were presented by Daniel Daigle in [2], [3] and [4]. For this, we introduce basic fundamenta1 results of the theory of derivations in a ring and we present results on locally nilpotents derivations in a domain with characteristic zero and unique factorization. One of these results is the Jacobian forrnula that we use to describe the set of the equivalent loca11y nilpotents derivations of B = k[x, y, z] and the set LND(B) where B = k[x, y]. Moreover, we give equivalent conditions to the existence of a ?-homogeneous locally nilpotent derivation in the ring B = k[x, y, z] with kernel k[¿, g], {¿} and {g} e B, and mdc(?) = mdc(?(¿), ? (g)) = 1
Mestrado
Algebra
Mestre em Matemática
34
Gokhale, Dhananjay R. "Resolutions mod I, Golod pairs." Diss., Virginia Tech, 1992. http://hdl.handle.net/10919/39431.
Full textAbstract:
Let R be a commutative ring, I be an ideal in R and let M be a R/ I -module. In this thesis we construct a R/ I -projective resolution of M using given R-projective resolutions of M and I. As immediate consequences of our construction we give descriptions of the canonical maps ExtR/I(M,N) -> ExtR(M,N) and TorRN(M, N) -> TorR/In(M, N) for a R/I module N and we give a new proof of a theorem of Gulliksen [6] which states that if I is generated by a regular sequence of length r then ∐∞n=o TorR/In (M, N) is a graded module over the polynomial ring R/ I [X₁. .. Xr] with deg Xi = -2, 1 ≤ i ≤ r. If I is generated by a regular element and if the R-projective dimension of M is finite, we show that M has a R/ I-projective resolution which is eventually periodic of period two.
This generalizes a result of Eisenbud [3]. In the case when R = (R, m) is a Noetherian local ring and M is a finitely generated R/ I -module, we discuss the minimality of the constructed resolution. If it is minimal we call (M, I) a Golod pair over R. We give a direct proof of a theorem of Levin [10] which states thdt if (M,I) is a Golod pair over R then (ΩnR/IR/I(M),I) is a Golod pair over R where ΩnR/IR/I(M) is the nth syzygy of the constructed R/ I -projective resolution of M. We show that the converse of the last theorem is not true and if (Ω¹R/IR/I(M),I) is a Golod pair over R then we give a necessary and sufficient condition for (M, I) to be a Golod pair over R.
Finally we prove that if (M, I) is a Golod pair over R and if a ∈ I - mI is a regular element in R then (M, (a)) and (1/(a), (a)) are Golod pairs over R and (M,I/(a)) is a Golod pair over R/(a). As a corrolary of this result we show that if the natural map π : R → R/1 is a Golod homomorphism ( this means (R/m, I) is a Golod pair over R ,Levin [8]), then the natural maps π₁ : R → R/(a) and π₂ : R/(a) → R/1 are Golod homomorphisms.Ph. D.
35
Berni, Jean Cerqueira. "Some algebraic and logical aspects of C∞-Rings." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/45/45131/tde-14022019-203839/.
Full textAbstract:
As pointed out by I. Moerdijk and G. Reyes in [63], C∞-rings have been studied specially for their use in Singularity Theory and in order to construct topos models for Synthetic Differential Geometry. In this work, we follow a complementary trail, deepening our knowledge about them through a more pure bias, making use of Category Theory and accounting them from a logical-categorial viewpoint. We begin by giving a comprehensive systematization of the fundamental facts of the (equational) theory of C∞-rings, widespread here and there in the current literature - mostly without proof - which underly the theory of C∞-rings. Next we develop some topics of what we call a ∞Commutative Algebra, expanding some partial results of [66] and [67]. We make a systematic study of von Neumann-regular C∞-rings (following [2]) and we present some interesting results about them, together with their (functorial) relationship with Boolean spaces. We study some sheaf theoretic notions on C∞-rings, such as ∞(locally)-ringed spaces and the smooth Zariski site. Finally we describe classifying toposes for the (algebraic) theory of ∞ rings, the (coherent) theory of local C∞-rings and the (algebraic) theory of von Neumann regular C∞-rings.Conforme observado por I. Moerdijk e G. Reyes em [63], os anéis C∞ têm sido estudados especialmente tendo em vista suas aplicações em Teoria de Singularidades e para construir toposes que sirvam de modelos para a Geometria Diferencial Sintética. Neste trabalho, seguimos um caminho complementar, aprofundando nosso conhecimento sobre eles por um viés mais puro, fazendo uso da Teoria das Categorias e os analisando a partir de pontos de vista algébrico e lógico-categorial. Iniciamos o trabalho apresentando uma sistematização abrangente dos fatos fundamentais da teoria (equacional) dos anéis C∞, distribuídos aqui e ali na literatura atual - a maioria sem demonstrações - mas que servem de base para a teoria. Na sequência, desenvolvemos alguns tópicos do que denominamos Álgebra Comutativa C∞, expandindo resultados parciais de [66] e [67]. Realizamos um estudo sistemático dos anéis C∞ von Neumann-regulares - na linha do estudo algébrico realizado em [2]- e apresentamos alguns resultados interessantes a seu respeito, juntamente com sua relação (funtorial) com os espaços booleanos. Estudamos algumas noções pertinentes à Teoria de Feixes para anéis ∞, tais como espaços (localmente) ∞anelados e o sítio de Zariski liso. Finalmente, descrevemos toposes classicantes para a teoria (algébrica) dos anéis C∞, a teoria (coerente) dos anéis locais C∞ e a teoria (algébrica) dos anéis C∞ von Neumann regulares.
36
Lavila, Vidal Olga. "On the diagonals of a Rees algebra." Doctoral thesis, Universitat de Barcelona, 1999. http://hdl.handle.net/10803/53578.
Full textAbstract:
The aim of this work is to study the ring-theoretic properties of the diagonals of a Rees algebra, which from a geometric point of view are the homogenous coordinate rings of embeddings of blow-ups of projective varieties along a subvariety. First we are going to introduce the subject and the main problems. After that we shall review the known results about these problems, and finally we will give a summary of the contents and results obtained in this work.L’objectiu d’aquesta memòria és l’estudi de les propietats aritmètiques de les diagonals d’una àlgebra de Rees o, des d’un punt de vista geomètric, dels anells de coordenades homogenis d’immersions d’explosions de varietats projectives al llarg d’una subvarietat. En primer lloc, anem a introduir el tema i els principals problemes que tractarem. A continuació, exposarem els resultats coneguts sobre aquests problemes i finalment farem un resum dels resultats obtinguts en aquesta memòria.
37
Duncan, A. J. "Two topics in commutative ring theory." Thesis, University of Edinburgh, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.234124.
Full text
38
Hedenlund, Alice. "Galois Theory of Commutative Ring Spectra." Thesis, KTH, Matematik (Avd.), 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-183512.
Full textAbstract:
This thesis discusses Galois theory of ring spectra in the sense of John Rognes. The aim is to give a clear introduction that provides a solid foundation for further studies into the subject. We introduce ring spectra using the symmetric spectra of Hovey, Shipley and Smith, and discuss the symmetric monoidal model structure on this category. We define and give results for Galois extensions of these objects. We also give examples involving Eilenberg-Mac Lane spectra of commutative rings, topological K-theory spectra and cochain algebras of these. Galois extensions of ring spectra are compared to Ga-lois extensions of commutative rings especially relating to faithfulness, a property that is implicit in the latter, but not in the former. This is proven by looking at extensions of cochain algebras using Eilenberg-Mac Lane spectra. We end by contrasting this to cochain algebra extensions using K-theory spectra, and show that such extensions are not Galois, using methods of Baker and Richter.Denna uppsats behandlar Galoisutvidgningar av ringspektra som först introducerade av Rognes. Målet är att ge en klar introduktion för en sta-bil grund för vidare studier inom ämnet. Vi introducerar ringspektra genom att använda oss av symmetris-ka spektra utvecklade av Hovey, Shipley och Smith, och diskuterar den symmetriskt monoidiala modelstrukturen på denna kategori. Vi definierar och ger resultat för Galoisutvidgningar av dessa objekt. Vi ger också en mängd exempel, som till exempel utvidgningar av Eilenberg-Mac Lane spektra av kommutativa ringar, topologiska K-teorispektra och koked-jealgebror. Galoisutvidgningar av ringspektra jämförs med Galoisutvidgningar av kommutativa ringar, speciellt med avseende pa˚ trogenhet, en egenskap som ¨ar en inneboende egenskap hos den senare men inte i den förra. Detta visas genom att betrakta utvidgningar av kokedjealgebror av Eilenberg-Mac Lane spektra. Vi avslutar med att jämföra detta med kokedjealgebrautvidgningar av K-teorispektra och visar att sådana inte är Galois genom att använda metoder utvecklade av Baker och Richter
39
Ferreira, Mauricio de Araujo 1982. "Algebras biquaternionicas : construção, classificação e condições de existencia via formas quadraticas e involuções." [s.n.], 2006. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306541.
Full textAbstract:
Orientador: Antonio Jose EnglerDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
Made available in DSpace on 2018-08-05T18:56:31Z (GMT). No. of bitstreams: 1 Ferreira_MauriciodeAraujo_M.pdf: 1033477 bytes, checksum: 8d697b5cdeb1a633c1270a5e2f919de7 (MD5) Previous issue date: 2006
Resumo: Neste trabalho, estudamos as álgebras biquaterniônicas, que são um tipo especial de álgebra central simples de dimensão 16, obtida como produto tensorial de duas álgebras de quatérnios. A teoria de formas quadráticas é aplicada para estudarmos critérios de decisão sobre quando uma álgebra biquaterniônica é de divisão e quando duas destas álgebras são isomorfas. Além disso, utilizamos o u-invariante do corpo para discutirmos a existência de álgebras biquaterniônicas de divisão sobre o corpo. Provamos também um resultado atribuído a A. A. Albert, que estabelece critérios para decidir quando uma álgebra central simples de dimensão 16 é de fato uma álgebra biquaterniônica, através do estudo de involuções. Ao longo do trabalho, construímos vários exemplos concretos de álgebras biquaterniônicas satisfazendo propriedades importantes
Mestrado
Algebra
Mestre em Matemática
40
Stalvey, Harrison. "Weak Primary Decomposition of Modules Over a Commutative Ring." Digital Archive @ GSU, 2010. http://digitalarchive.gsu.edu/math_theses/84.
Full textAbstract:
This paper presents the theory of weak primary decomposition of modules over a commutative ring. A generalization of the classic well-known theory of primary decomposition, weak primary decomposition is a consequence of the notions of weakly associated prime ideals and nearly nilpotent elements, which were introduced by N. Bourbaki. We begin by discussing basic facts about classic primary decomposition. Then we prove the results on weak primary decomposition, which are parallel to the classic case. Lastly, we define and generalize the Compatibility property of primary decomposition.
41
Johnston, Ann. "Markov Bases for Noncommutative Harmonic Analysis of Partially Ranked Data." Scholarship @ Claremont, 2011. http://scholarship.claremont.edu/hmc_theses/4.
Full textAbstract:
Given the result $v_0$ of a survey and a nested collection of summary statistics that could be used to describe that result, it is natural to ask which of these summary statistics best describe $v_0$. In 1998 Diaconis and Sturmfels presented an approach for determining the conditional significance of a higher order statistic, after sampling a space conditioned on the value of a lower order statistic. Their approach involves the computation of a Markov basis, followed by the use of a Markov process with stationary hypergeometric distribution to generate a sample.This technique for data analysis has become an accepted tool of algebraic statistics, particularly for the study of fully ranked data. In this thesis, we explore the extension of this technique for data analysis to the study of partially ranked data, focusing on data from surveys in which participants are asked to identify their top $k$ choices of $n$ items. Before we move on to our own data analysis, though, we present a thorough discussion of the Diaconis–Sturmfels algorithm and its use in data analysis. In this discussion, we attempt to collect together all of the background on Markov bases, Markov proceses, Gröbner bases, implicitization theory, and elimination theory, that is necessary for a full understanding of this approach to data analysis.
42
Alshaniafi, Y. S. "The homological grade of a module over a commutative ring." Thesis, University of Southampton, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.280830.
Full text
43
Baig, Muslim. "Primary Decomposition and Secondary Representation of Modules over a Commutative Ring." Digital Archive @ GSU, 2009. http://digitalarchive.gsu.edu/math_theses/69.
Full textAbstract:
This paper presents the theory of Secondary Representation of modules over a commutative ring and their Attached Primes; introduced in 1973 by I. MacDonald as a dual to the important theory of associated primes and primary decomposition in commutative algebra. The paper explores many of the basic aspects of the theory of primary decomposition and associated primes of modules in the hopes to delineate and motivate the construction of a secondary representation, when possible. The thesis discusses the results of the uniqueness of representable modules and their attached primes, and, in particular, the existence of a secondary representation for Artinian modules. It concludes with some interesting examples of both secondary and representable modules, highlighting the consequences of the results thus established.
44
Martini, Lorenzo. "Local coherence of hearts in the derived category of a commutative ring." Doctoral thesis, Università degli studi di Trento, 2022. http://hdl.handle.net/11572/354322.
Full textAbstract:
Approximation theory is a fundamental tool in order to study the representation theory of a ring R. Roughly speaking, it consists in determining suitable additive or abelian subcategories of the whole module category Mod-R with nice enough functorial properties. For example, torsion theory is a well suited incarnation of approximation theory. Of course, such an idea has been generalised to the additive setting itself, so that both Mod-R and other interesting categories related with R may be linked functorially. By the seminal work of Beilinson, Bernstein and Deligne (1982), the derived category of the ring turns out to admit useful torsion theories, called t-structures: they are pairs of full subcategories of D(R) whose intersection, called the heart, is always an abelian category. The so-called standard t-structure of D(R) has as its heart the module category Mod-R itself. Since then a lot of results devoted to the module theoretic characterisation of the hearts have been achieved, providing evidence of the usefulness of the t-structures in the representation theory of R. In 2020, following a research line promoted by many other authors, Saorin and Stovicek proved that the heart of any compactly generated t-structure is always a locally finitely presented Grothendieck categories (actually, this is true for any t-structure in a triangulated category with coproducts). Essentially, this means that the hearts of D(R) come equipped with a finiteness condition miming that one valid in Mod-R. In the present thesis we tackle the problem of characterising when the hearts of certain compactly generated t-structures of a commutative ring are even locally coherent. In this commutative context, after the works of Neeman and Alonso, Jeremias and Saorin, compactly generated t-structures turned out to be very interesting over a noetherian ring, for they are in bijection with the Thomason filtrations of the prime spectrum. In other words, they are classified by geometric objects, moreover their constituent subcategories have a precise cohomological description. However, if the ascending chain condition lacks, such classification is somehow partial, though provided by Hrbek. The crucial point is that the constituents of the t-structures have a different description w.r.t. that available in the noetherian setting, yet if one copies the latter for an arbitrary ring still obtains a t-structure, but it is not clear whether it must be compactly generated. Consequently, pursuing the study of the local coherence of the hearts given by a Thomason filtration, we ended by considering two t-structures. Our technique in order to face the lack of the ascending chain condition relies on a further approximation of the hearts by means of suitable torsion theories. The main results of the thesis are the following: we prove that for the so-called weakly bounded below Thomason filtrations the two t-structures have the same heart (therefore it is always locally finitely presented), and we show that they coincide if and only they are both compactly generated. Moreover, we achieve a complete characterisation of the local coherence for the hearts of the Thomason filtrations of finite length.
45
Philippoussis, Anthony. "Necessary and sufficient conditions so that a commutative ring can be embedded into a strongly [pi]-regular ring." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape15/PQDD_0007/MQ39934.pdf.
Full text
46
Nossem, Nicole. "On the perfect closure of a commutative Noetherian ring of positive prime characteristic." Thesis, University of Sheffield, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.251466.
Full text
47
Crawford, Simon Philip. "Singularities of noncommutative surfaces." Thesis, University of Edinburgh, 2018. http://hdl.handle.net/1842/31543.
Full textAbstract:
The primary objects of study in this thesis are noncommutative surfaces; that is, noncommutative noetherian domains of GK dimension 2. Frequently these rings will also be singular, in the sense that they have infinite global dimension. Very little is known about singularities of noncommutative rings, particularly those which are not finite over their centre. In this thesis, we are able to give a precise description of the singularities of a few families of examples. In many examples, we lay the foundations of noncommutative singularity theory by giving a precise description of the singularities of the fundamental examples of noncommutative surfaces. We draw comparisons with the fundamental examples of commutative surface singularities, called Kleinian singularities, which arise from the action of a finite subgroup of SL(2; k) acting on a polynomial ring. The main tool we use to study the singularities of noncommutative surfaces is the singularity category, first introduced by Buchweitz in [Buc86]. This takes a (possibly noncommutative) ring R and produces a triangulated category Dsg(R) which provides a measure of "how singular" R is. Roughly speaking, the size of this category reflects how bad the singularity is; in particular, Dsg(R) is trivial if and only if R has finite global dimension. In [CBH98], Crawley-Boevey-Holland introduced a family of noncommutative rings which can be thought of as deformations of the coordinate ring of a Kleinian singularity. We give a precise description of the singularity categories of these deformations, and show that their singularities can be thought of as unions of (commutative) Kleinian singularities. In particular, our results show that deforming a singularity in this setting makes it no worse. Another family of noncommutative surfaces were introduced by Rogalski-Sierra-Stafford in [RSS15b]. The authors showed that these rings share a number of ring-theoretic properties with deformations of type A Kleinian singularities. We apply our techniques to show that the "least singular" example has an A1 singularity, and conjecture that other examples exhibit similar behaviour. In [CKWZ16a], Chan-Kirkman-Walton-Zhang gave a definition for a quantum version of Kleinian singularities. These require the data of a two-dimensional AS regular algebra A and a finite group G acting on A with trivial homological determinant. We extend a number of results in [CBH98] to the setting of quantum Kleinian singularities. More precisely, we show that one can construct deformations of the skew group rings A#G and the invariant rings AG, and then determine some of their ring-theoretic properties. These results allow us to give a precise description of the singularity categories of quantum Kleinian singularities, which often have very different behaviour to their non-quantum analogues.
48
Nachar, Georges. "Caténarité et anneaux de séries formelles." Lyon 1, 1985. http://www.theses.fr/1985LYO11673.
Full textAbstract:
Soient r un anneau commutatif de dimension de krull finie et k son corps des fractions. On note r(x) (resp. R((x))) l'anneau des polynomes (resp. Des series formelles) en x sur r. J. T. Arnold (1) a montre que si r n'est pas un sft-anneau alors dim r((x))=infini. De plus, si d est un sft-anneau prueferien, dim d((x::(1),. . . ,x::(n)))=n dim d+1. D'autre part, pour un anneau noetherien d, dim d((x::(1),. . . ,x::(n)))= dim d+n. S. Doering et y. Lequain ont montre que les proprietes suivantes sont equivalentes : a) d(x::(1),. . . ,x::(n)) est catenaire; b) d((x::(1),. . . ,x::(n))) est catenaire; c) pour tout ideal premier p de d, d::(p)((x::(1),. . . ,xn)) est catenaire. On tente, dans ce travail, d'etendre cette propriete au cas d'un sft-anneau prueferien. Pour un ideal premier p d'un anneau de valuation discrete v. De dimension finie (son groupe des ordres est isomorphe a z**(m)), on met en evidence une famille denombrable d'elements de v engendrant p. De plus on montre que pour un anneau de valuation v, dim v >ou= 2, les deux proprietes suivantes sont equivalentes : 1) v(1x::(1),. . . ,x::(n))) est un anneau catenaire; 2) v est un anneau de valuation discrete (sft-anneau de valuation) et n=1. On etend ce resultat au cas des sft-anneaux prueferiens. On acheve ce travail en donnant un exemple de sft-anneau integre t de la forme t=d+m qui n'est ni noetherien ni prueferien et qui verifie les proprietes suivantes : dim t((x))=dim t+1; 2) t((x)) est un anneau catenaire
49
Mbirika, Abukuse III. "Analysis of symmetric function ideals: towards a combinatorial description of the cohomology ring of Hessenberg varieties." Diss., University of Iowa, 2010. https://ir.uiowa.edu/etd/708.
Full textAbstract:
Symmetric functions arise in many areas of mathematics including combinatorics, topology and algebraic geometry. Using ideals of symmetric functions, we tie these three branches together. This thesis generalizes work of Garsia and Procesi in 1992 that gave a quotient ring presentation for the cohomology ring of Springer varieties.
Let R be the polynomial ring Ζ[x1,…,xn]. We present two different ideals in R. Both are parametrized by a Hessenberg function h, namely a nondecreasing function that satisfies h(i) ≥ i for all i. The first ideal, which we call Ih, is generated by modified elementary symmetric functions. The ideal I_h generalizes the work of Tanisaki who gave a combinatorial description of the ideal used in Garsia and Procesi's quotient ring. Like the Tanisaki ideal, the generating set for Ih is redundant. We give a minimal generating set for this ideal. The second ideal, which we call Jh, is generated by modified complete symmetric functions. The generators of this ideal form a Gröbner basis, which is a useful property. Using the Gröbner basis for Jh, we identify a basis for the quotient R/Jh.
We introduce a partial ordering on the Hessenberg functions, and in turn we discover nice nesting properties in both families of ideals. When h>h', we have Ih ⊂ Ih' and Jh ⊂ Jh'. We prove that Ih equals Jh when h is maximal. Since Ih is the ideal generated by the elementary symmetric functions when h is maximal, the generating set for Jh forms a Gröbner basis for the elementary symmetric functions. Moreover, the quotient R/Jh gives another description of the cohomology ring of the full flag variety.
The generators of the ring R/Jh are in bijective correspondence with the Betti numbers of certain Hessenberg varieties. These varieties are a two-parameter generalization of Springer varieties, parametrized by a nilpotent operator X and a Hessenberg function h. These varieties were introduced in 1992 by De Mari, Procesi and Shayman. We provide evidence that as h varies, the quotient R/Jh may be a presentation for the cohomology ring of a subclass of Hessenberg varieties called regular nilpotent varieties.
50
Chaitou, Mohamad. "Performance of multicast packet aggregation with quality of service support in all-optical packet-switched ring networks." Evry, Institut national des télécommunications, 2006. http://www.theses.fr/2006TELE0007.
Full textAbstract:
La technologie optique WDM (Wavelength Division Multiplexing) subit un grand succès grâce aux besoins croissants en debit du traffic basé sur le protocole IP (Inter- net Protocol). Une solution efficace pour l'adaptation du traffic IP au niveau WDM dans les reseaux metropolitains optiques (Metropolitan Area Networks-MANs) consiste µa l'utilisation de la commutation de paquets optiques (Optical Packet Switching-OPS), basee sur le multiplexage statistique des paquets optiques de taille fixe dans un environnement synchrone. Ce travail presente un nouveau mecanisme effcace pour supporter directement le traffc IP dans les reseaux WDM tout en respectant différents critrères de quallité de service (Quality of Service-QoS). L'approche consiste à agreger les paquets IP independamment de leurs destinations finales, dans un seul paquet optique de taille fixe qui sera efffectivement un paquet optique multicast. Le support de QoS est assuré par l'utilisation d'un temporisateur et par la repartition du traffc IP en plusieurs classes de service (Class of Service-CoS) dans les noeuds de bord du reseau optique (edge nodes). Après agregation, les paquets optiques construits appartiendront à la même classe de service ce qui simplifie le plan de contrôle du reseau optique. La complexite et la différentiation de services sont implementees dans les noeuds de bord. Nous quantifions la performance de plusieurs versions de la technique d'agrégation par plusieurs modèles analytiques, valides par des simulations. Chaque version correspond à un fonctionnement différent du temporisateur ou à la possibilité de segmentation ou pas d'un paquet IP dans le processus d'agrégation. Ceci est dû au fait que les paquets IP ont une taille variable, tandis que le paquet optique a une taille fixe. Les modèles analytiques proposés tiennent compte de cette variabilité ce qui présente une contribution importante de ce travail. Nous explorons parsuite l'influence du multicast sur les réseaux métropolitains ayant une architecture en anneau. Nous analysons en particulier deux familles de réseaux. La première est celle qui emploie la technique "destination stripping" et la deuxième emploie la technique "hub stripping". Nous développons des modèles analytiques pour évaluer la capacité, et les delais d'accès, et nous validons ces modèles par des simulations qui montrent aussi l'effet du tracé sporadique sur les résultats. Nous démontrons l'avantage de la technique d'agrégation multicast dans le MAN en la comparant au cas d'agrégation unicast et au cas où il n'y a pas d'agrégation. L'agrégation multicast offre une bonne utilisation de la bande passante sans introduire aucune complèxite dans le cas de la technique "hub stripping". En outre, une petite addition au niveau des informations de contrôle est nécessaire dans le cas du "destination stripping"The explosive growth of the Internet Protocol (IP)-based traffic has accelerated the emergence of wavelength division multiplexing (WDM) technology. In order to provide a simple transport platform of IP traffic over WDM structure, optical packet switching (OPS), based on fixed-length packets and synchronous node operation, is regarded as a long term issue especially for metropolitan area networks (MANs) where the synchronization is easy to handle and relatively simple to maintain. In this context, this thesis presents a novel approach for efficiently supporting IP traffic with several quality of service (QoS) requirements into a synchronous WDM MAN layer. The claimed efficiency is achieved by aggregating IP packets regard-less of their final destinations which yields a multicast optical packet. To support QoS, a timer mechanism is used and a class-based scheme at the edge of optical network is adopted. Several analytical models have been developed to quantify the performance of different versions of the aggregation approach. The different versions correspond to different timer mechanisms and to the permission or the ban of IP packet segmentation by the aggregation process. This is because IP packets have variable size while optical packet is of fixed size. The length variability of IP traffic is included in the analytical models which represents an essential contribution of this thesis. We next investigate the impact of multicast on WDM slotted ring MANs. In particular we focus on two families of MANs. The first family enables destination stripping, while in the second one, ring nodes contain passive components and the stripping is attributed to the hub which separates two different sets of wave-length channels, one for transmission and one for reception. The capacity (maximum throughput) of each network is evaluated by means of an analytical model. The access delay is also investigated by using an approximate approach in the case of destination stripping and an exact approach in the case of hub stripping. Further-more, the impact of the optical packet format on the performance is depicted. We show the attractiveness of the multicast aggregation in MANs through a comparative study with the performance of unicast aggregation and no aggregation approaches therein. Multicast aggregation increases bandwidth efficiency due to the filling ratio improvement exhibited in optical packets. Furthermore, hub stripping networks match very well the multicast nature of the generated optical packets without the addition of any complexity in the node architecture. However, a small overhead complexity is added in the case of destination stripping networks. Note that all approximative analytical models have been validated by using extensive simulations, where two traffic profiles were investigated: Poisson and Self-Similar