Ideal Bases and Primary Decomposition : Case of Two Variables
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Noncommutative Unique Factorization Domainso
NONCOMMUTATIVE UNIQUE FACTORIZATION DOMAINSO BY P. M. COHN 1. Introduction. By a (commutative) unique factorization domain (UFD) one usually understands an integral domain R (with a unit-element) satisfying the following three conditions (cf. e.g. Zariski-Samuel [16]): Al. Every element of R which is neither zero nor a unit is a product of primes. A2. Any two prime factorizations of a given element have the same number of factors. A3. The primes occurring in any factorization of a are completely deter- mined by a, except for their order and for multiplication by units. If R* denotes the semigroup of nonzero elements of R and U is the group of units, then the classes of associated elements form a semigroup R* / U, and A1-3 are equivalent to B. The semigroup R*jU is free commutative. One may generalize the notion of UFD to noncommutative rings by taking either A-l3 or B as starting point. It is obvious how to do this in case B, although the class of rings obtained is rather narrow and does not even include all the commutative UFD's. This is indicated briefly in §7, where examples are also given of noncommutative rings satisfying the definition. However, our principal aim is to give a definition of a noncommutative UFD which includes the commutative case. Here it is better to start from A1-3; in order to find the precise form which such a definition should take we consider the simplest case, that of noncommutative principal ideal domains. For these rings one obtains a unique factorization theorem simply by reinterpreting the Jordan- Holder theorem for right .R-modules on one generator (cf. -
The History of the Formulation of Ideal Theory
The History of the Formulation of Ideal Theory Reeve Garrett November 28, 2017 1 Using complex numbers to solve Diophantine equations From the time of Diophantus (3rd century AD) to the present, the topic of Diophantine equations (that is, polynomial equations in 2 or more variables in which only integer solutions are sought after and studied) has been considered enormously important to the progress of mathematics. In fact, in the year 1900, David Hilbert designated the construction of an algorithm to determine the existence of integer solutions to a general Diophantine equation as one of his \Millenium Problems"; in 1970, the combined work (spanning 21 years) of Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson showed that no such algorithm exists. One such equation that proved to be of interest to mathematicians for centuries was the \Bachet equa- tion": x2 +k = y3, named after the 17th century mathematician who studied it. The general solution (for all values of k) eluded mathematicians until 1968, when Alan Baker presented the framework for constructing a general solution. However, before this full solution, Euler made some headway with some specific examples in the 18th century, specifically by the utilization of complex numbers. Example 1.1 Consider the equation x2 + 2 = y3. (5; 3) and (−5; 3) are easy to find solutions, but it's 2 not obviousp whetherp or not there are others or whatp they might be. Euler realized by factoring x + 2 as (x + 2i)(x − 2i) and then using the facts that Z[ 2i] is a UFD andp the factorsp given are relatively prime that the solutions above are the only solutions,p namelyp because (x + 2i)(x − 2i) being a cube forces each of these factors to be a cube (i.e. -
Assignment 2, Due 10/09 Before Class Begins
ASSIGNMENT 2, DUE 10/09 BEFORE CLASS BEGINS RANKEYA DATTA You may talk to each other while attempting the assignment, but you must write up your own solutions, preferably on TeX. You must also list the names of all your collaborators. You may not use resources such as MathStackExchange and MathOverflow. Please put some effort into solving the assignments since they will often develop important theory I will use in class. The assignments have to be uploaded on Gradescope. Please note that Gradescope will not accept late submissions. All rings are commutative. Problem 1. In this problem you will construct a non-noetherian ring. Consider the group G := Z ⊕ Z; with a total ordering as follows: (a1; b1) < (a2; b2) if and only if either a1 < a2 or a1 = a2 and b1 < b2. Here total ordering means that any two elements of G are comparable. Define a function ν : C(X; Y ) − f0g ! G P α β as follows: for a polynomial f = (α,β) cαβX Y 2 C[X; Y ] − f0g, ν(f) := minf(α; β): cαβ 6= 0g: For an arbitrary element f=g 2 C(X; Y ) − f0g, where f; g are polynomials, ν(f=g) := ν(f) − ν(g): (1) Show that ν is a well-defined group homomorphism. (2) Show that for all x; y 2 C(X; Y ) − f0g such that x + y 6= 0, ν(x + y) ≥ minfν(x); ν(y)g. (3) Show that Rν := fx 2 C(X; Y )−f0g : ν(x) ≥ (0; 0)g[f0g is a subring of C(X; Y ) containing C[X; Y ]. -
Generalized Brauer Dimension of Semi-Global Fields
GENERALIZED BRAUER DIMENSION OF SEMI-GLOBAL FIELDS by SAURABH GOSAVI A dissertation submitted to the School of Graduate Studies Rutgers, The State University of New Jersey In partial fulfillment of the requirements For the degree of Doctor of Philosophy Graduate Program in Department of Mathematics Written under the direction of Daniel Krashen and approved by New Brunswick, New Jersey October, 2020 ABSTRACT OF THE DISSERTATION Generalized Brauer Dimension Of Semi-global Fields By Saurabh Gosavi Dissertation Director: Daniel Krashen Let F be a one variable function field over a complete discretely valued field with residue field k. Let n be a positive integer, coprime to the characteristic of k. Given a finite subgroup B in the n-torsion part of the Brauer group n Br F , we define the index of B to be the minimum of the degrees of field extensions which( ) split all elements in B. In this thesis, we improve an upper bound for the index of B, given by Parimala-Suresh, in terms of arithmetic invariants of k and k t . As a simple application of our result, given a quadratic form q F , where F is a function( ) field in one variable over an m-local field, we provide an upper-bound~ to the minimum of degrees of field extensions L F so that the Witt index of q L becomes the largest possible. ~ ⊗ ii Acknowledgements I consider myself fortunate to have met a number of people without whom this thesis would not have been possible. I am filled with a profound sense of gratitude towards my thesis advisor Prof. -
Commutative Algebra
Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions . -
Commutative Ideal Theory Without Finiteness Conditions: Completely Irreducible Ideals
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 358, Number 7, Pages 3113–3131 S 0002-9947(06)03815-3 Article electronically published on March 1, 2006 COMMUTATIVE IDEAL THEORY WITHOUT FINITENESS CONDITIONS: COMPLETELY IRREDUCIBLE IDEALS LASZLO FUCHS, WILLIAM HEINZER, AND BRUCE OLBERDING Abstract. An ideal of a ring is completely irreducible if it is not the intersec- tion of any set of proper overideals. We investigate the structure of completely irrreducible ideals in a commutative ring without finiteness conditions. It is known that every ideal of a ring is an intersection of completely irreducible ideals. We characterize in several ways those ideals that admit a representation as an irredundant intersection of completely irreducible ideals, and we study the question of uniqueness of such representations. We characterize those com- mutative rings in which every ideal is an irredundant intersection of completely irreducible ideals. Introduction Let R denote throughout a commutative ring with 1. An ideal of R is called irreducible if it is not the intersection of two proper overideals; it is called completely irreducible if it is not the intersection of any set of proper overideals. Our goal in this paper is to examine the structure of completely irreducible ideals of a commutative ring on which there are imposed no finiteness conditions. Other recent papers that address the structure and ideal theory of rings without finiteness conditions include [3], [4], [8], [10], [14], [15], [16], [19], [25], [26]. AproperidealA of R is completely irreducible if and only if there is an element x ∈ R such that A is maximal with respect to not containing x. -
Eigenschemes and the Jordan Canonical Form
EIGENSCHEMES AND THE JORDAN CANONICAL FORM HIROTACHI ABO, DAVID EKLUND, THOMAS KAHLE, AND CHRIS PETERSON ABSTRACT. We study the eigenscheme of a matrix which encodes information about the eigenvectors and generalized eigenvectors of a square matrix. The two main results in this paper are a decomposition of the eigenscheme of a matrix into primary components and the fact that this decomposition encodes the numeric data of the Jordan canonical form of the matrix. We also describe how the eigenscheme can be interpreted as the zero locus of a global section of the tangent bundle on projective space. This interpretation allows one to see eigenvectors and generalized eigenvectors of matrices from an alternative viewpoint. CONTENTS 1. Introduction 1 2. Eigenvectors and eigenschemes 4 3. IdealsofJordanmatriceswithasingleeigenvalue 6 4. Ideals of general Jordan matrices 15 5. Eigenschemes and tangent bundles 20 6. The discriminant 21 References 22 1. INTRODUCTION Motivation. Let K be a field about which we make no a priori assumptions. For any r r r r r × matrix A K × , a non-zero vector v K is an eigenvector if Av is a scalar multiple of v or equivalently∈ if Av and v are linearly∈ dependent. The span of v is a one dimensional subspace of Kr and its projectivization can thus be viewed as a point in the projectivization of Kr. In other words, the subspace of Kr spanned by an eigenvector v of A determines a point [v] Pr 1. Let R = K[x ,..., x ], let x be the column vector of variables, and ∈ − 1 r arXiv:1506.08257v3 [math.AG] 12 Apr 2016 let (Ax x) be the r 2 matrix obtained by concatenating the two r 1 matrices Ax and x side by| side. -
On Primary Factorizations”
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 54 (1988) l-11-154 141 North-Holland ON PRIMARY FACTORIZATIONS” D.D. ANDERSON Department qf Mathematics, University qf lowa, Iowa City, IA 52242, U.S.A. L.A. MAHANEY ** Department of Mathematics, Dallas Baptist Unrversity, Dallas, TX 7521 I, U.S.A. Communicated by C.A. Wcibel Received 11 February 1987 We relate ideals in commutative rings which are products of primary ideals to ideals with primary decompositions. Invertible primary ideals are shown to have special properties. Suffi- cient conditions are given for a primary product ideal to habe a unique product representation. A domain is weakly factorial if every non-unit i5 a product of primary elements. If R is weakly factorial, Pit(R) = 0. A Noctherian weakly factorial domain R is factorial precisely when R i\ ill- tegrally closed. R[X] is weakly factorial if and only if R is a weakly factorial GCD domain. Pro- perties of weakly factorial GCD domains are discussed. 1. Introduction Throughout this paper all rings will be assumed to be commutative with identity. We first discuss ideals that are a product of primary ideals and relate primary pro- ducts to primary decompositions. An ideal with a primary decomposition need not be a product of primary ideals and an ideal that is a product of primary ideals need not have a primary decomposition. However, we show that if an ideal generated by an R-sequence is a product of primary ideals, then it has a primary decomposition. -
Algebraic Geometry Over Groups I: Algebraic Sets and Ideal Theory
Algebraic geometry over groups I: Algebraic sets and ideal theory Gilbert Baumslag Alexei Myasnikov Vladimir Remeslennikov Abstract The object of this paper, which is the ¯rst in a series of three, is to lay the foundations of the theory of ideals and algebraic sets over groups. TABLE OF CONTENTS 1. INTRODUCTION 1.1 Some general comments 1.2 The category of G-groups 1.3 Notions from commutative algebra 1.4 Separation and discrimination 1.5 Ideals 1.6 The a±ne geometry of G-groups 1.7 Ideals of algebraic sets 1.8 The Zariski topology of equationally Noetherian groups 1.9 Decomposition theorems 1.10 The Nullstellensatz 1.11 Connections with representation theory 1.12 Related work 2. NOTIONS FROM COMMUTATIVE ALGEBRA 2.1 Domains 2.2 Equationally Noetherian groups 2.3 Separation and discrimination 2.4 Universal groups 1 3. AFFINE ALGEBRAIC SETS 3.1 Elementary properties of algebraic sets and the Zariski topology 3.2 Ideals of algebraic sets 3.3 Morphisms of algebraic sets 3.4 Coordinate groups 3.5 Equivalence of the categories of a±ne algebraic sets and coordinate groups 3.6 The Zariski topology of equationally Noetherian groups 4. IDEALS 4.1 Maximal ideals 4.2 Radicals 4.3 Irreducible and prime ideals 4.4 Decomposition theorems for ideals 5. COORDINATE GROUPS 5.1 Abstract characterization of coordinate groups 5.2 Coordinate groups of irreducible algebraic sets 5.3 Decomposition theorems for coordinate groups 6. THE NULLSTELLENSATZ 7. BIBLIOGRAPHY 2 1 Introduction 1.1 Some general comments The beginning of classical algebraic geometry is concerned with the geometry of curves and their higher dimensional analogues. -
Primary Decomposition
+ Primary Decomposition The decomposition of an ideal into primary ideals is a traditional pillar of ideal theory. It provides the algebraic foundation for decomposing an algebraic variety into its irreducible components-although it is only fair to point out that the algebraic picture is more complicated than naive geometry would suggest. From another point of view primary decomposition provides a gen- eralization of the factorization of an integer as a product of prime-powers. In the modern treatment, with its emphasis on localization, primary decomposition is no longer such a central tool in the theory. It is still, however, of interest in itself and in this chapter we establish the classical uniqueness theorems. The prototypes of commutative rings are z and the ring of polynomials kfxr,..., x,] where k is a field; both these areunique factorization domains. This is not true of arbitrary commutative rings, even if they are integral domains (the classical example is the ring Z[\/=1, in which the element 6 has two essentially distinct factorizations, 2.3 and it + r/-S;(l - /=)). However, there is a generalized form of "unique factorization" of ideals (not of elements) in a wide class of rings (the Noetherian rings). A prime ideal in a ring A is in some sense a generalization of a prime num- ber. The corresponding generalization of a power of a prime number is a primary ideal. An ideal q in a ring A is primary if q * A and. if xy€q => eitherxeqory" eqforsomen > 0. In other words, q is primary o Alq * 0 and every zero-divisor in l/q is nilpotent. -
Computing Limit Points of Quasi-Components of Regular Chains and Its Applications
Western University Scholarship@Western Electronic Thesis and Dissertation Repository 5-11-2017 12:00 AM Computing Limit Points of Quasi-components of Regular Chains and its Applications Parisa Alvandi The University of Western Ontario Supervisor Dr. Marc Moreno Maza The University of Western Ontario Graduate Program in Computer Science A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Parisa Alvandi 2017 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Theory and Algorithms Commons Recommended Citation Alvandi, Parisa, "Computing Limit Points of Quasi-components of Regular Chains and its Applications" (2017). Electronic Thesis and Dissertation Repository. 4565. https://ir.lib.uwo.ca/etd/4565 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Abstract Computing limits is a fundamental task in mathematics and different mathematical con- cepts are defined in terms of limit computations. Among these mathematical concepts, we are interested in three different types of limit computations: first, computing the limit points of so- lutions of polynomial systems represented by regular chains, second, computing tangent cones of space curves at their singular points which can be viewed as computing limit of secant lines, and third, computing the limit of real multivariate rational functions. For computing the limit of solutions of polynomial systems represented by regular chains, we present two different methods based on Puiseux series expansions and linear changes of coordinates. -
Ideals and Class Groups of Number Fields
Ideals and class groups of number fields A thesis submitted To Kent State University in partial Fulfillment of the requirements for the Degree of Master of Science by Minjiao Yang August, 2018 ○C Copyright All rights reserved Except for previously published materials Thesis written by Minjiao Yang B.S., Kent State University, 2015 M.S., Kent State University, 2018 Approved by Gang Yu , Advisor Andrew Tonge , Chair, Department of Mathematics Science James L. Blank , Dean, College of Arts and Science TABLE OF CONTENTS…………………………………………………………...….…...iii ACKNOWLEDGEMENTS…………………………………………………………....…...iv CHAPTER I. Introduction…………………………………………………………………......1 II. Algebraic Numbers and Integers……………………………………………......3 III. Rings of Integers…………………………………………………………….......9 Some basic properties…………………………………………………………...9 Factorization of algebraic integers and the unit group………………….……....13 Quadratic integers……………………………………………………………….17 IV. Ideals……..………………………………………………………………….......21 A review of ideals of commutative……………………………………………...21 Ideal theory of integer ring 풪푘…………………………………………………..22 V. Ideal class group and class number………………………………………….......28 Finiteness of 퐶푙푘………………………………………………………………....29 The Minkowski bound…………………………………………………………...32 Further remarks…………………………………………………………………..34 BIBLIOGRAPHY…………………………………………………………….………….......36 iii ACKNOWLEDGEMENTS I want to thank my advisor Dr. Gang Yu who has been very supportive, patient and encouraging throughout this tremendous and enchanting experience. Also, I want to thank my thesis committee members Dr. Ulrike Vorhauer and Dr. Stephen Gagola who help me correct mistakes I made in my thesis and provided many helpful advices. iv Chapter 1. Introduction Algebraic number theory is a branch of number theory which leads the way in the world of mathematics. It uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Concepts and results in algebraic number theory are very important in learning mathematics.