DOCSLIB.ORG

IDEAL THEORY of LOCAL QUADRATIC TRANSFORMS of REGULAR LOCAL RINGS a Dissertation Submitted to the Faculty of Purdue University B

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
IDEAL THEORY of LOCAL QUADRATIC TRANSFORMS of REGULAR LOCAL RINGS a Dissertation Submitted to the Faculty of Purdue University B IDEAL THEORY OF LOCAL QUADRATIC TRANSFORMS OF REGULAR LOCAL RINGS A Dissertation Submitted to the Faculty of Purdue University by Matthew J. Toeniskoetter In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2017 Purdue University West Lafayette, Indiana ii THE PURDUE UNIVERSITY GRADUATE SCHOOL STATEMENT OF DISSERTATION APPROVAL Dr. William Heinzer, Chair Department of Mathematics Dr. Bernd Ulrich Department of Mathematics Dr. Giulio Caviglia Department of Mathematics Dr. Edray Goins Department of Mathematics Approved by: Dr. David Goldberg Head of the School Graduate Program iii ACKNOWLEDGMENTS First and foremost, I would like to thank my advisor, Professor William Heinzer, from whom I've learned a great deal. I look back fondly at all the time spent in his office, in classrooms around campus, and in the campus dining halls discussing mathematics. I appreciate all of the support and encouragement he has given me over the years, and while I have not always followed it, I am grateful for all of the helpful advice he has offered. I am thankful to my committee members Professors Bernd Ulrich and Giulio Caviglia, whose graduate level courses have been essential in my understanding of advanced topics in commutative algebra. I am grateful for all of the opportunities to speak at the weekly commutative algebra seminar, which have been invaluable experiences. I am grateful to have Professor Edray Goins on my thesis committee. I thank him for his help both in preparing my thesis defense and in editing and formatting this document. I am also thankful for the opportunity to TA for his research experience for undergraduates program in the summer of my third year in graduate school. I would like to thank my collaborators Youngsu Kim, Mee-Kyoung Kim, Alan Loper, Guerrieri Lorenzo, Bruce Olberding, and Hans Schoutens. I had a great time working with all of you. In addition, I would like to thank Alan for the invitation to speak at the Ohio State University commutative algebra seminar, and Bruce, for his contributions to the project that comprises much of my thesis. I am thankful to the undergraduate professors involved in the research for under- graduates program at Oakland University: Eddie Cheng, Meir Shillor, and especially Anna Spagnuolo, who sent me on the path to Purdue University. I would like to thank the mathematics department at Purdue University for pro- viding me with the wonderful opportunity to pursue the study of mathematics. iv Finally, I thank my parents. I am grateful for all of their patience and support throughout my study of mathematics. Matthew Toeniskoetter West Lafayette, July 19, 2017 v TABLE OF CONTENTS Page ABSTRACT ::::::::::::::::::::::::::::::::::::: vii 1 INTRODUCTION :::::::::::::::::::::::::::::::: 1 2 PRELIMINARIES :::::::::::::::::::::::::::::::: 7 2.1 Notation and Definitions :::::::::::::::::::::::::: 7 2.2 The Dimension Formula :::::::::::::::::::::::::: 7 2.3 Ordered Abelian Groups :::::::::::::::::::::::::: 8 2.4 Valuations and Valuation Rings :::::::::::::::::::::: 9 2.5 Transforms of Ideals :::::::::::::::::::::::::::: 11 3 CONVERGENCE OF VALUATIONS :::::::::::::::::::::: 13 3.1 Valuation Rings as a Topological Space :::::::::::::::::: 13 3.2 Limit of Valuation Rings :::::::::::::::::::::::::: 14 3.3 Possibly Infinite Valuations :::::::::::::::::::::::: 15 4 SEQUENCES OF LOCAL QUADRATIC TRANSFORMS OF REGULAR LOCAL RINGS :::::::::::::::::::::::::::::::::: 18 4.1 Blow-Ups and Local Quadratic Transforms :::::::::::::::: 18 4.2 Local Quadratic Transforms and Transforms of Ideals :::::::::: 20 4.3 Quadratic Shannon Extensions :::::::::::::::::::::: 21 4.4 The Noetherian Hull :::::::::::::::::::::::::::: 25 4.5 The Boundary Valuation :::::::::::::::::::::::::: 26 5 THE STRUCTURE OF A SHANNON EXTENSION ::::::::::::: 28 5.1 The Intersection Decomposition :::::::::::::::::::::: 29 5.2 The Asymptotic Limit of Order Valuations :::::::::::::::: 30 5.3 The Limit of Transforms :::::::::::::::::::::::::: 33 5.4 The Transform Formula for w ::::::::::::::::::::::: 34 5.5 The Archimedean Case ::::::::::::::::::::::::::: 37 5.6 The Complete Integral Closure :::::::::::::::::::::: 41 5.7 The Non-Archimedean Case :::::::::::::::::::::::: 43 5.8 Completely Integrally Closed Examples :::::::::::::::::: 46 6 LIMITS OF REES VALUATIONS ::::::::::::::::::::::: 50 6.1 Normalizing Local Quadratic Transforms ::::::::::::::::: 50 6.2 Rees Valuations ::::::::::::::::::::::::::::::: 51 6.3 Multiplicity and Blowing Up :::::::::::::::::::::::: 53 6.4 Degree Functions :::::::::::::::::::::::::::::: 55 vi Page 6.5 Convergence of Rees Valuations :::::::::::::::::::::: 57 7 MONOMIAL LOCAL QUADRATIC TRANSFORMS ::::::::::::: 60 7.1 Monomial Valuations :::::::::::::::::::::::::::: 61 7.2 Conditions for the union to be a valuation :::::::::::::::: 63 REFERENCES :::::::::::::::::::::::::::::::::::: 68 vii ABSTRACT Toeniskoetter, Matthew J. PhD, Purdue University, August 2017. Ideal Theory of Local Quadratic Transforms of Regular Local Rings. Major Professor: William Heinzer. Let R be a regular local ring of dimension d ≥ 2. To a non-divisorial valuation V that dominates R, there is an associated infinite sequence of local quadratic transforms fRngn≥0 of R along V . Abhyankar has shown that if d = 2, then the union S = S n≥0 Rn is equal to V , and in higher dimensions, Shannon and Granja et al. have given equivalent conditions that the union S equals V . In this thesis, we examine properties of the ring S in the case where S is not equal to V . We associate to S a minimal proper Noetherian overring, called the Noetherian hull. Each Rn has an associated order valuation ordRn , and we show that the sequence of order valuations fordRn gn≥0 converges to a valuation called the boundary valuation. We show that S is the intersection of its Noetherian hull and boundary valuation ring, and we go on to study these rings in detail. This naturally breaks down into an archimedean and a non-archimedean case, and for each case, we construct an explicit description for the boundary valuation. Then, after loosening the condition that R is regular and replacing the sequence fordRn gn≥0 with the sequence of Rees valuations fνngn≥0 of the maximal ideals of Rn, we prove an analogous result about the convergence of Rees valuations. 1 1. INTRODUCTION The notion of blowing up is a powerful tool that arises in the study of commutative algebra and algebraic geometry. Speaking loosely and geometrically, the notion of blowing up is to replace a subspace of a space with a larger space that can distinguish the different directions through the original subspace. For instance, the curve x2 − y2 + x3 = 0 takes two distinct paths through the origin. By blowing up the origin in the plane, we replace the origin with a projective line. Then after taking the strict transform of the curve to the blown-up space, the two distinct paths that the curve takes through the origin now intersect this projective line in two distinct points and the curve no longer intersects itself. Algebraically, to blow-up the ideal I of the Noetherian ring R, one uses the Rees algebra of I, defined as M R[It] = Intn = R ⊕ It ⊕ I2t2 ⊕ · · · : n≥0 The blow-up of I is the projective scheme ProjR[It]. Now let R be a regular local ring of dimension d ≥ 2 and let m = (x1; : : : ; xd) be the maximal ideal of R. Take the blow-up of m, look at the local ring of a point in the fiber of m, and call this ring R1. The ring R1 is a local quadratic transform of R. Since mR1 is principal, mR1 = xkR1 for some k with 1 ≤ k ≤ d. Algebraically, the ring R1 has the form x1 xd R1 = R ; ··· ; xk xk n for some prime ideal n such that xk 2 n. It is well-known that the ring R1 is itself a regular local ring and the dimension of R1 is at most the dimension of R. One may iterate this process, and in this way, obtain a sequence f(Rn; mn)gn≥0 of local quadratic transforms of regular local rings. The dimensions of the rings in this sequence are non-increasing positive integers, so the dimension must stabilize. 2 This process of iterating local quadratic transforms of the same dimension corre- sponds to the geometric notion of tracking nonsingular closed points through repeated blow-ups. We can think of the ring R itself as some nonsingular closed point on some variety. In taking a local quadratic transform, one blows up the closed point R on this variety, then chooses a closed point R1 in the fiber of R. An instance where such a sequence naturally arises is through a valuation ring V that birationally dominates R = R0. The valuation ring V has a unique center on the blow-up ProjR[mt] of m, which uniquely determines the local quadratic transform of R along V . This iterative geometric process is a powerful tool, especially in a two-dimensional setting. It plays a central role in the embedded resolution of singularities for curves on surfaces (see, for example, [2] and [5, Sections 3.4 and 3.5]), as well as factorization of birational morphisms between nonsingular surfaces (see [1, Theorem 3] and [28, Lemma, p. 538]). Algebraically, iterated local quadratic transforms are an essential component in Zariski's theory of integrally closed ideals in two-dimensional regular local rings (see [29, Appendix 5]). Every such ideal has a unique factorization into irreducible such ideals, and the irreducible ideals are in one-to-one correspondence with the two-dimensional regular local rings that can be attained through iterated local quadratic transforms. Abhyankar proves that for a sequence fRngn≥0 of iterated local quadratic trans- S forms of two-dimensional regular local rings, the union Rn is a valuation ring [1, n≥0 Lemma 12]. For rings of higher dimension, this is no longer true. Shannon gives ex- S amples [27, Examples 4.7 and 4.17] that show that the union S = Rn of a sequence n≥0 of local quadratic transforms of three-dimensional regular local rings need not be a valuation ring. In the same paper, Shannon proves an equivalent condition for the union S to be a rank 1 valuation ring.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Open Problems in Commutative Ring Theory
    Open problems in commutative ring theory Paul-Jean Cahen , Marco Fontana y, Sophie Frisch zand Sarah Glaz x December 23, 2013 Abstract This article consists of a collection of open problems in commuta- tive algebra. The collection covers a wide range of topics from both Noetherian and non-Noetherian ring theory and exhibits a variety of re- search approaches, including the use of homological algebra, ring theoretic methods, and star and semistar operation techniques. The problems were contributed by the authors and editors of this volume, as well as other researchers in the area. Keywords: Prüfer ring, homological dimensions, integral closure, group ring, grade, complete ring, McCoy ring, Straight domain, di- vided domain, integer valued polynomials, factorial, density, matrix ring, overring, absorbing ideal, Kronecker function ring, stable ring, divisorial domain, Mori domain, finite character, PvMD, semistar operation, star operation, Jaffard domain, locally tame domain, fac- torization, spectrum of a ring, integral closure of an ideal, Rees al- gebra, Rees valuation. Mthematics Subject Classification (2010): 13-02; 13A05; 13A15; 13A18; 13B22; 13C15; 13D05; 13D99; 13E05; 13F05; 13F20; 13F30; 13G05 1 Introduction This article consists of a collection of open problems in commutative algebra. The collection covers a wide range of topics from both Noetherian and non- Noetherian ring theory and exhibits a variety of research approaches, including Paul-Jean Cahen (Corresponding author), 12 Traverse du Lavoir de Grand-Mère, 13100 Aix en Provence, France. e-mail: [email protected] yMarco Fontana, Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Largo San Leonardo Murialdo 1, 00146 Roma, Italy. e-mail: [email protected] zSophie Frisch, Mathematics Department, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria.
    [Show full text]
  • New Constructive Methods in Classical Ideal Theory
    JOURNALOF ALGEBRA 100, 138-178 (1986) New Constructive Methods in Classical Ideal Theory H. MICHAEL MILLER FB Mathematik & Informatik, FernUniversitiit, D 5800 Hagen I, Federal Republic of Germany AND FERDINANDOMORA Istituto di Matematica, Universitri di Genova, Genova, Italy Communicated by D. A. Buchsbaum Received August 10, 1983 The recent development of Computer Algebra allows us to take up problems of classical Ideal Theory, for which constructive solutions by nearly unfeasible algorithms exist like the problem of deciding whether a polynomial belongs to a given ideal (G. Hermann, Math. Ann. 95 (1926) 736-788). Even Macaulay’s methods by using H-bases are not satisfactory, at least when considering modules. The concept of Grobner bases, introduced in Computer Algebra by Buchberger, allows us to find feasible algorithms for many problems of classical Ideal Theory by transforming them into some low dimensional systems of linear equations. Our aim is to demonstrate the effectivity of this new concept. Considering only polynomial rings P := k[X,,..., X.1, k a field, we generalize in Part I the notion of Grijbner bases to modules, present some equivalent properties characterizing these bases and show their good computational qualities. In Part II we construct starting with Grobner bases of ideals explicit free solutions, where all the bases of submodules under consideration are also Grobner bases, and present a reduction strategy for obtaining shorter resolutions, where all the bases have also favourable properties (T-bases). Part III deals with applications. We propose a method for constructing minimal resolutions starting with the resolutions of Part II and present a con- venient way for computing the Hilbert function.
    [Show full text]
  • The Arithmetic Structure of Discrete Dynamical Systems on the Torus
    The arithmetic structure of discrete dynamical systems on the torus Dissertation zur Erlangung des Doktorgrades der Mathematik an der Fakultät für Mathematik der Universität Bielefeld vorgelegt von Natascha Neumärker Bielefeld, im Mai 2012 Erstgutachter: Professor Dr. Michael Baake Zweitgutachter: Professor John A.G. Roberts, PhD Gedruckt auf alterungsbeständigem Papier nach ISO 9706. Contents 1 Introduction 1 2 Preliminaries 5 2.1 Thetorus ....................................... 5 2.2 The rational lattices and related rings, modules and groups ........... 5 2.3 Integermatrices................................. ... 6 2.4 Orbit counts and generating functions . ........ 7 3 Locally invertible toral endomorphisms on the rational lattices 9 3.1 Somereductions .................................. 9 3.2 Subgroups and submodules induced by integer matrices . ........... 10 3.3 Local versus global orbit counts of toral endomorphisms ............. 11 3.4 Matrix order on lattices and periods of points . .......... 14 3.5 Powers of integer matrices . ..... 16 3.6 Results from the theory of linear recursions . .......... 17 3.7 Results for d = 2 ................................... 20 3.8 Normal forms and conjugacy invariants . ........ 23 4 Orbit pretail structure of toral endomorphisms 30 4.1 Generalstructure................................ 30 4.2 Thepretailtree.................................. 31 4.3 Decomposition and parametrisation on Λ˜ pr .................... 37 4.4 Classification on Λ˜ p .................................. 40 4.5 Sequences of pretail trees and the ‘global’ pretail tree ............... 42 5 Symmetry and reversibility 46 5.1 Reversibility of SL(2, Z)-matrices mod n ...................... 47 5.2 Reversibility in GL(2, Fp) .............................. 48 5.3 Reversibility mod n .................................. 52 5.4 Matrix order and symmetries over Fp ........................ 53 6 The Casati-Prosen map on rational lattices of the torus 56 6.1 Reversibility and symmetric orbits .
    [Show full text]
  • Ideal Bases and Primary Decomposition : Case of Two Variables
    J. Symbolic Computation (1985) 1, 261-270 Ideal Bases and Primary Decomposition : Case of Two Variables D . LAZARDt Institut de Programmation, Universite Paris VI, F-75230 Paris Cedex 05 (Received 24 September 1984) A complete structure theorem is given for standard (= Grobner) bases for bivariate polynomials over a field and lexicographical orderings or for univariate polynomials over a Euclidian ring . An easy computation of primary decomposition in such rings is deduced . Another consequence is a natural factorisation of the resultant of two univariate polynomials over the integers which is a generalisation of the "reduced discriminant" of a polynomial of degree 2 . 1. Introduction In his celebrated paper, Hironaka (1964), introduced the notion of a standard basis of an ideal of formal power series . One year later, Buchberger (1965) defined independently the same notion for polynomial ideals, which he termed Grobner basis, and gave an algorithm for computing such a basis . He proved also that the roots of a system of algebraic equations can easily be deduced from a Grobner basis relative to a lexicographical ordering . In this paper we use Hironaka's terminology of standard basis for the general notion and Buchberger's term of Grobner basis for the particular case of lexicographical orderings . There are now many papers on this subject (see Buchberger (1985) for a large bibliography and also EUROSAM 84 and EUROCAL 85 for more recent references) . It appears that the computational complexity of standard bases is strongly dependent on the ordering of monomials and on the irregularity of the ideal. However, once a standard basis is computed, most "properties" of an ideal can be easily (i.e .
    [Show full text]
  • Prime and Primary Ideals
    MATHEMATICS PRIME AND PRIMARY IDEALS BY L. C. A. VAN LEEUWEN (Communicated by Prof. H. FREUDENTHAL at the meeting of October 28, 1961) Introduction In a previous paper [5] we have tried to extend a theory of ideals in commutative rings to the non-commutative case. In this paper we investigate further properties of the concepts which were introduced in [5]. It turns out that much of KRULL's work in [4] can be extended to non­ commutative rings. In his paper [3] Krull also treated two-sided ideals in non-commutative rings, but our point of view is different from that Df Krull in [3]. With the introduction of isolated primary components of an ideal a (theorem 5) we needed an assumption about the isolated Z-components of a in order to prove that these ideals are primary; namely that every minimal prime ideal belonging to the isolated Z-component ideal i~ of a (,P is a given minimal prime ideal of a) is Z-related to i~ . For the definitions of these and other concepts we refer to our previous paper (5]. An example shows that our condition is not fulfilled in every ring. For commutative rings it is proved by McCoY [9, Theorem 27] for an arbitrary ideal a. In the non-commutative case sufficient conditions that the isolated [-component ideal i~ of a be primary are obtained by supposing that i~ has only a finite number of minimal prime ideals. If now the radical m(i~) (in the sense of McCoy) is represented by an irredundant representation of minimal primes ,Pt (i.e.
    [Show full text]
  • Commutative Ideal Theory Without Finiteness Conditions: Primal Ideals
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9947(XX)0000-0 COMMUTATIVE IDEAL THEORY WITHOUT FINITENESS CONDITIONS: PRIMAL IDEALS LASZLO FUCHS, WILLIAM HEINZER, AND BRUCE OLBERDING Abstract. Our goal is to establish an efficient decomposition of an ideal A of a commutative ring R as an intersection of primalT ideals. We prove the existence of a canonical primal decomposition: A = A ,wherethe P ∈XA (P ) A(P ) are isolated components of A that are primal ideals having distinct and incomparable adjoint primes P . For this purpose we define the set Ass(A) of associated primes of the ideal A to be those defined and studied by Krull. We determine conditions for the canonical primal decomposition to be irre- dundant, or residually maximal, or the unique representation of A as an irre- dundant intersection of isolated components of A. Using our canonical primal decomposition, we obtain an affirmative answer to a question raised by Fuchs in [5] and also prove for P ∈ Spec R that an ideal A ⊆ P is an intersection of P -primal ideals if and only if the elements of R \ P are prime to A.We prove that the following conditions are equivalent: (i) the ring R is arith- metical, (ii) every primal ideal of R is irreducible, (iii) each proper ideal of R is an intersection of its irreducible isolated components. We classify the rings for which the canonical primal decomposition of each proper ideal is an irredundant decomposition of irreducible ideals as precisely the arithmetical rings with Noetherian maximal spectrum.
    [Show full text]
  • NOTES on IDEALS 1. Introduction Let R Be a Commutative Ring
    NOTES ON IDEALS KEITH CONRAD 1. Introduction Let R be a commutative ring (with identity). An ideal in R is an additive subgroup I ⊂ R such that for all x 2 I, Rx ⊂ I. Example 1.1. For a 2 R, (a) := Ra = fra : r 2 Rg is an ideal. An ideal of the form (a) is called a principal ideal with generator a. We have b 2 (a) if and only if a j b. Note (1) = R. An ideal containing an invertible element u also contains u−1u = 1 and thus contains every r 2 R since r = r · 1, so the ideal is R. This is why (1) = R is called the unit ideal: it's the only ideal containing units (invertible elements). Example 1.2. For a and b 2 R, (a; b) := Ra + Rb = ra + r0b : r; r0 2 R is an ideal. It is called the ideal generated by a and b. More generally, for a1; : : : ; an 2 R the set (a1; a2; : : : ; an) = Ra1 + ··· + Ran is an ideal in R, called a finitely generated ideal or the ideal generated by a1; : : : ; an. In some rings every ideal is principal, or more broadly every ideal is finitely generated, but there are also some \big" rings in which some ideal is not finitely generated. It would be wrong to say an ideal is non-principal if it is described with two generators: an ideal generated by several elements might be generated by fewer elements and even by one element (a principal ideal). For example, in Z, (1.1) (6; 8) = 6Z + 8Z =! 2Z: Both 8 and 6 are elements of the ideal (6; 8), so 8 − 6 = 2 is in the ideal.
    [Show full text]
  • Progress in Commutative Algebra 2
    Progress in Commutative Algebra 2 Progress in Commutative Algebra 2 Closures, Finiteness and Factorization edited by Christopher Francisco Lee Klingler Sean Sather-Wagstaff Janet C. Vassilev De Gruyter Mathematics Subject Classification 2010 13D02, 13D40, 05E40, 13D45, 13D22, 13H10, 13A35, 13A15, 13A05, 13B22, 13F15 An electronic version of this book is freely available, thanks to the support of libra- ries working with Knowledge Unlatched. KU is a collaborative initiative designed to make high quality books Open Access. More information about the initiative can be found at www.knowledgeunlatched.org This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 License. For details go to http://creativecommons.org/licenses/by-nc-nd/4.0/. ISBN 978-3-11-027859-0 e-ISBN 978-3-11-027860-6 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. ” 2012 Walter de Gruyter GmbH & Co. KG, Berlin/Boston Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing: Hubert & Co. GmbH & Co. KG, Göttingen ϱ Printed on acid-free paper Printed in Germany www.degruyter.com Preface This collection of papers in commutative algebra stemmed out of the 2009 Fall South- eastern American Mathematical Society Meeting which contained three special ses- sions in the field: Special Session on Commutative Ring Theory, a Tribute to the Memory of James Brewer, organized by Alan Loper and Lee Klingler; Special Session on Homological Aspects of Module Theory, organized by Andy Kustin, Sean Sather-Wagstaff, and Janet Vassilev; and Special Session on Graded Resolutions, organized by Chris Francisco and Irena Peeva.
    [Show full text]