Commutative Algebra
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Multiplicative Theory of Ideals This Is Volume 43 in PURE and APPLIED MATHEMATICS a Series of Monographs and Textbooks Editors: PAULA
Multiplicative Theory of Ideals This is Volume 43 in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: PAULA. SMITHAND SAMUELEILENBERG A complete list of titles in this series appears at the end of this volume MULTIPLICATIVE THEORY OF IDEALS MAX D. LARSEN / PAUL J. McCARTHY University of Nebraska University of Kansas Lincoln, Nebraska Lawrence, Kansas @ A CADEM I C P RE S S New York and London 1971 COPYRIGHT 0 1971, BY ACADEMICPRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS. ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London WlX 6BA LIBRARY OF CONGRESS CATALOG CARD NUMBER: 72-137621 AMS (MOS)1970 Subject Classification 13F05; 13A05,13B20, 13C15,13E05,13F20 PRINTED IN THE UNITED STATES OF AMERICA To Lillie and Jean This Page Intentionally Left Blank Contents Preface xi ... Prerequisites Xlll Chapter I. Modules 1 Rings and Modules 1 2 Chain Conditions 8 3 Direct Sums 12 4 Tensor Products 15 5 Flat Modules 21 Exercises 27 Chapter II. Primary Decompositions and Noetherian Rings 1 Operations on Ideals and Submodules 36 2 Primary Submodules 39 3 Noetherian Rings 44 4 Uniqueness Results for Primary Decompositions 48 Exercises 52 Chapter Ill. Rings and Modules of Quotients 1 Definition 61 2 Extension and Contraction of Ideals 66 3 Properties of Rings of Quotients 71 Exercises 74 Vii Vlll CONTENTS Chapter IV. -
Commutative Algebra, Lecture Notes
COMMUTATIVE ALGEBRA, LECTURE NOTES P. SOSNA Contents 1. Very brief introduction 2 2. Rings and Ideals 2 3. Modules 10 3.1. Tensor product of modules 15 3.2. Flatness 18 3.3. Algebras 21 4. Localisation 22 4.1. Local properties 25 5. Chain conditions, Noetherian and Artin rings 29 5.1. Noetherian rings and modules 31 5.2. Artin rings 36 6. Primary decomposition 38 6.1. Primary decompositions in Noetherian rings 42 6.2. Application to Artin rings 43 6.3. Some geometry 44 6.4. Associated primes 45 7. Ring extensions 49 7.1. More geometry 56 8. Dimension theory 57 8.1. Regular rings 66 9. Homological methods 68 9.1. Recollections 68 9.2. Global dimension 71 9.3. Regular sequences, global dimension and regular rings 76 10. Differentials 83 10.1. Construction and some properties 83 10.2. Connection to regularity 88 11. Appendix: Exercises 91 References 100 1 2 P. SOSNA 1. Very brief introduction These are notes for a lecture (14 weeks, 2×90 minutes per week) held at the University of Hamburg in the winter semester 2014/2015. The goal is to introduce and study some basic concepts from commutative algebra which are indispensable in, for instance, algebraic geometry. There are many references for the subject, some of them are in the bibliography. In Sections 2-8 I mostly closely follow [2], sometimes rearranging the order in which the results are presented, sometimes omitting results and sometimes giving statements which are missing in [2]. In Section 9 I mostly rely on [9], while most of the material in Section 10 closely follows [4]. -
Nilpotent Elements Control the Structure of a Module
Nilpotent elements control the structure of a module David Ssevviiri Department of Mathematics Makerere University, P.O BOX 7062, Kampala Uganda E-mail: [email protected], [email protected] Abstract A relationship between nilpotency and primeness in a module is investigated. Reduced modules are expressed as sums of prime modules. It is shown that presence of nilpotent module elements inhibits a module from possessing good structural properties. A general form is given of an example used in literature to distinguish: 1) completely prime modules from prime modules, 2) classical prime modules from classical completely prime modules, and 3) a module which satisfies the complete radical formula from one which is neither 2-primal nor satisfies the radical formula. Keywords: Semisimple module; Reduced module; Nil module; K¨othe conjecture; Com- pletely prime module; Prime module; and Reduced ring. MSC 2010 Mathematics Subject Classification: 16D70, 16D60, 16S90 1 Introduction Primeness and nilpotency are closely related and well studied notions for rings. We give instances that highlight this relationship. In a commutative ring, the set of all nilpotent elements coincides with the intersection of all its prime ideals - henceforth called the prime radical. A popular class of rings, called 2-primal rings (first defined in [8] and later studied in [23, 26, 27, 28] among others), is defined by requiring that in a not necessarily commutative ring, the set of all nilpotent elements coincides with the prime radical. In an arbitrary ring, Levitzki showed that the set of all strongly nilpotent elements coincides arXiv:1812.04320v1 [math.RA] 11 Dec 2018 with the intersection of all prime ideals, [29, Theorem 2.6]. -
Classifying the Representation Type of Infinitesimal Blocks of Category
Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J. Platt (Under the Direction of Brian D. Boe) Abstract Let g be a simple Lie algebra over the field C of complex numbers, with root system Φ relative to a fixed maximal toral subalgebra h. Let S be a subset of the simple roots ∆ of Φ, which determines a standard parabolic subalgebra of g. Fix an integral weight ∗ µ µ ∈ h , with singular set J ⊆ ∆. We determine when an infinitesimal block O(g,S,J) := OS of parabolic category OS is nonzero using the theory of nilpotent orbits. We extend work of Futorny-Nakano-Pollack, Br¨ustle-K¨onig-Mazorchuk, and Boe-Nakano toward classifying the representation type of the nonzero infinitesimal blocks of category OS by considering arbitrary sets S and J, and observe a strong connection between the theory of nilpotent orbits and the representation type of the infinitesimal blocks. We classify certain infinitesimal blocks of category OS including all the semisimple infinitesimal blocks in type An, and all of the infinitesimal blocks for F4 and G2. Index words: Category O; Representation type; Verma modules Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J. Platt B.A., Utah State University, 1999 M.S., Utah State University, 2001 M.A, The University of Georgia, 2006 A Thesis Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Athens, Georgia 2008 c 2008 Kenyon J. Platt All Rights Reserved Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J. -
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. -
Hyperbolicity of Hermitian Forms Over Biquaternion Algebras
HYPERBOLICITY OF HERMITIAN FORMS OVER BIQUATERNION ALGEBRAS NIKITA A. KARPENKO Abstract. We show that a non-hyperbolic hermitian form over a biquaternion algebra over a field of characteristic 6= 2 remains non-hyperbolic over a generic splitting field of the algebra. Contents 1. Introduction 1 2. Notation 2 3. Krull-Schmidt principle 3 4. Splitting off a motivic summand 5 5. Rost correspondences 7 6. Motivic decompositions of some isotropic varieties 12 7. Proof of Main Theorem 14 References 16 1. Introduction Throughout this note (besides of x3 and x4) F is a field of characteristic 6= 2. The basic reference for the staff related to involutions on central simple algebras is [12].p The degree deg A of a (finite dimensional) central simple F -algebra A is the integer dimF A; the index ind A of A is the degree of a central division algebra Brauer-equivalent to A. Conjecture 1.1. Let A be a central simple F -algebra endowed with an orthogonal invo- lution σ. If σ becomes hyperbolic over the function field of the Severi-Brauer variety of A, then σ is hyperbolic (over F ). In a stronger version of Conjecture 1.1, each of two words \hyperbolic" is replaced by \isotropic", cf. [10, Conjecture 5.2]. Here is the complete list of indices ind A and coindices coind A = deg A= ind A of A for which Conjecture 1.1 is known (over an arbitrary field of characteristic 6= 2), given in the chronological order: • ind A = 1 | trivial; Date: January 2008. Key words and phrases. -
Ring (Mathematics) 1 Ring (Mathematics)
Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right. -
CATENARITY in MODULE-FINITE ALGEBRAS 1. Introduction Let R Be
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 12, Pages 3495{3502 S 0002-9939(99)04962-X Article electronically published on May 13, 1999 CATENARITY IN MODULE-FINITE ALGEBRAS SHIRO GOTO AND KENJI NISHIDA (Communicated by Ken Goodearl) Abstract. The main theorem says that any module-finite (but not necessarily commutative) algebra Λ over a commutative Noetherian universally catenary ring R is catenary. Hence the ring Λ is catenary if R is Cohen-Macaulay. When R is local and Λ is a Cohen-Macaulay R-module, we have that Λ is a catenary ring, dim Λ = dim Λ=Q +htΛQfor any Q Spec Λ, and the equality ∈ n =htΛQ htΛP holds true for any pair P Q of prime ideals in Λ and for − ⊆ any saturated chain P = P0 P1 Pn =Qof prime ideals between P ⊂ ⊂···⊂ and Q. 1. Introduction Let R be a commutative Noetherian ring and let Λ be an R-algebra which is finitely generated as an R-module. Here we don’t assume Λ to be a commutative ring. The purpose of this article is to prove the following Theorem 1.1. Any module-finite R-algebra Λ is catenary if R is universally cate- nary. Before going ahead, let us recall the definition of catenarity and universal cate- narity ([Ma, p. 84]). We say that our ring Λ is catenary if for any pair P Q of ⊆ prime ideals in Λ and for any saturated chain P = P0 P1 Pn = Q of prime ideals between P and Q, the length n of the chain⊂ is independent⊂ ··· ⊂ of its par- ticular choice and hence equal to htΛ=P Q=P ,wherehtΛ=P Q=P denotes the height of the prime ideal Q=P in Λ=P . -
The Pseudo-Radical of a Commutative Ring
Pacific Journal of Mathematics THE PSEUDO-RADICAL OF A COMMUTATIVE RING ROBERT WILLIAM GILMER,JR. Vol. 19, No. 2 June 1966 PACIFIC JOURNAL OF MATHEMATICS Vol. 19, No. 2, 1966 THE PSEUDO-RADICAL OF A COMMUTATIVE RING ROBERT W. GILMER, JR. If D is an integral domain with identity having quotient field K, the pseudo-radical of D is defined to be the inter- section of all nonzero prime ideals of D. Consideration of the pseudo-radical arises naturally in examining the relation between the statements "D has Jacobson radical zero" and "D[u] has Jacobson radical zero, where ueK". Theorem 4 proves that the first statement implies the second. As a corollary it follows that if M is a prime ideal of the polynomial ring R[X] over a commutative ring R and if P=MnR, then M is an intersection of maximal ideals of R[X] if P is an intersection of maximal ideals of R. Consequently, if R is a Hubert ring, R[X] is also a Hubert ring. The remainder of the paper is devoted to a study of domains having nonzero pseudo-radical. Goldman has defined in [6] the concept of a Hubert ring: the commutative ring R with identity is a Hubert ring if each proper prime ideal of R is an intersection of maximal ideals; here proper means an ideal different from R. The terminology is motivated by the observation that Hubert's Nullstellensatz may be interpreted as asserting that each proper prime ideal of the polynomial domain K[Xlf - —, Xn] for K a field, is an intersection of maximal ideals. -
SPECIALIZATION and INTEGRAL CLOSURE 3 Provided U = U
SPECIALIZATION AND INTEGRAL CLOSURE JOOYOUN HONG AND BERND ULRICH Abstract. We prove that the integral closedness of any ideal of height at least two is com- patible with specialization by a generic element. This opens the possibility for proofs using induction on the height of an ideal. Also, with additional assumptions, we show that an ele- ment is integral over a module if it is so modulo a generic element of the module. This turns questions about integral closures of modules into problems about integral closures of ideals, by means of a construction known as Bourbaki ideal. In this paper we prove in a rather general setting that the integral closure of ideals and modules is preserved under specialization modulo generic elements. Recall that the integral closure I of an ideal I in a commutative ring R is the set of all elements y that are integral over I, i.e., satisfy a polynomial equation of the form ym + α ym−1 + + α ym−i + + α = 0 , 1 · · · i · · · m where α Ii. Alternatively, one can consider the Rees algebra (I) of I, which is the i ∈ R subalgebra R[It] of the polynomial ring R[t]. The integral closure (I) of (I) in R[t] is again a graded algebra, and its graded components recover the integralR closuresR of all powers of I, (I)= R It I2t2 Iiti . R ⊕ ⊕ ⊕ ··· ⊕ ⊕ ··· In our main result, Theorem 2.1, we consider a Noetherian ring R such that the completion (Rm/√0) is reduced and equidimensional for every maximal ideal m of R; for instance, R could be an equidimensional excellent local ring. -
A Brief History of Ring Theory
A Brief History of Ring Theory by Kristen Pollock Abstract Algebra II, Math 442 Loyola College, Spring 2005 A Brief History of Ring Theory Kristen Pollock 2 1. Introduction In order to fully define and examine an abstract ring, this essay will follow a procedure that is unlike a typical algebra textbook. That is, rather than initially offering just definitions, relevant examples will first be supplied so that the origins of a ring and its components can be better understood. Of course, this is the path that history has taken so what better way to proceed? First, it is important to understand that the abstract ring concept emerged from not one, but two theories: commutative ring theory and noncommutative ring the- ory. These two theories originated in different problems, were developed by different people and flourished in different directions. Still, these theories have much in com- mon and together form the foundation of today's ring theory. Specifically, modern commutative ring theory has its roots in problems of algebraic number theory and algebraic geometry. On the other hand, noncommutative ring theory originated from an attempt to expand the complex numbers to a variety of hypercomplex number systems. 2. Noncommutative Rings We will begin with noncommutative ring theory and its main originating ex- ample: the quaternions. According to Israel Kleiner's article \The Genesis of the Abstract Ring Concept," [2]. these numbers, created by Hamilton in 1843, are of the form a + bi + cj + dk (a; b; c; d 2 R) where addition is through its components 2 2 2 and multiplication is subject to the relations i =pj = k = ijk = −1. -
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 .