DOCSLIB.ORG

Igor R. Shafarevich Schemes and Complex Manifolds Third Edition

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Igor R. Shafarevich Schemes and Complex Manifolds Third Edition Igor R. Shafarevich Basic Algebraic Geometry 2 Schemes and Complex Manifolds Third Edition Basic Algebraic Geometry 2 Igor R. Shafarevich Basic Algebraic Geometry 2 Schemes and Complex Manifolds Third Edition Igor R. Shafarevich Translator Algebra Section Miles Reid Steklov Mathematical Institute Mathematics Institute of the Russian Academy of Sciences University of Warwick Moscow, Russia Coventry, UK ISBN 978-3-642-38009-9 ISBN 978-3-642-38010-5 (eBook) DOI 10.1007/978-3-642-38010-5 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013945857 Mathematics Subject Classification (2010): 14-01 Translation of the 3rd Russian edition entitled “Osnovy algebraicheskoj geometrii”. MCCME, Moscow 2007, originally published in Russian in one volume © Springer-Verlag Berlin Heidelberg 1977, 1994, 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of pub- lication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface to Books 2–3 Books 2–3 correspond to Chapters V–IX of the first edition. They study schemes and complex manifolds, two notions that generalise in different directions the varieties in projective space studied in Book 1. Introducing them leads also to new results in the theory of projective varieties. For example, it is within the framework of the theory of schemes and abstract varieties that we find the natural proof of the adjunc- tion formula for the genus of a curve, which we have already stated and applied in Section 2.3, Chapter 4. The theory of complex analytic manifolds leads to the study of the topology of projective varieties over the field of complex numbers. For some questions it is only here that the natural and historical logic of the subject can be re- asserted; for example, differential forms were constructed in order to be integrated, a process which only makes sense for varieties over the (real or) complex fields. Changes from the First Edition As in the Book 1, there are a number of additions to the text, of which the following two are the most important. The first of these is a discussion of the notion of the algebraic variety classifying algebraic or geometric objects of some type. As an example we work out the theory of the Hilbert polynomial and the Hilbert scheme. I am very grateful to V.I. Danilov for a series of recommendations on this subject. In particular the proof of Theorem 6.7 is due to him. The second addition is the definition and basic properties of a Kähler metric, and a description (without proof) of Hodge’s theorem. Prerequisites Varieties in projective space will provide us with the main supply of examples, and the theoretical apparatus of Book 1 will be used, but by no means all of it. Differ- ent sections use different parts, and there is no point in giving exact indications. References to the Appendix are to the Algebraic Appendix at the end of Book 1. V VI Preface to Books 2–3 Prerequisites for the reader of Books 2–3 are as follows: for Book 2, the same as for Book 1; for Book 3, the definition of differentiable manifold, the basic theory of analytic functions of a complex variable, and a knowledge of homology, cohomol- ogy and differential forms (knowledge of the proofs is not essential); for Chapter 9, familiarity with the notion of fundamental group and the universal cover. References for these topics are given in the text. Recommendations for Further Reading For the reader wishing to go further in the study of algebraic geometry, we can recommend the following references. For the cohomology of algebraic coherent sheaves and their applications: see Hartshorne [37]. An elementary proof of the Riemann–Roch theorem for curves is given in W. Ful- ton, Algebraic curves. An introduction to algebraic geometry, W.A. Benjamin, Inc., New York–Amsterdam, 1969. This book is available as a free download from http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf. For the general case of Riemann–Roch, see A. Borel and J.-P. Serre, Le théorème de Riemann–Roch, Bull. Soc. Math. France 86 (1958) 97–136, Yu.I. Manin, Lectures on the K-functor in algebraic geometry, Uspehi Mat. Nauk 24:5 (149) (1969) 3–86, English translation: Russian Math. Surveys 24:5 (1969) 1–89, W. Fulton and S. Lang, Riemann–Roch algebra, Grundlehren der mathematis- chen Wissenschaften 277, Springer-Verlag, New York, 1985. Moscow, Russia I.R. Shafarevich Contents Book 2: Schemes and Varieties 5 Schemes .................................. 3 1 The Spec of a Ring .......................... 5 1.1 Definition of Spec A ..................... 5 1.2 Properties of Points of Spec A ................ 7 1.3 The Zariski Topology of Spec A ............... 9 1.4 Irreducibility, Dimension .................. 11 1.5 Exercises to Section 1 .................... 14 2 Sheaves . ............................. 15 2.1 Presheaves.......................... 15 2.2 The Structure Presheaf . ................... 17 2.3 Sheaves ........................... 19 2.4 Stalks of a Sheaf ....................... 23 2.5 Exercises to Section 2 .................... 24 3 Schemes . ............................. 25 3.1 Definition of a Scheme ................... 25 3.2 Glueing Schemes . ................... 30 3.3 Closed Subschemes . ................... 32 3.4 Reduced Schemes and Nilpotents .............. 35 3.5 Finiteness Conditions . ................... 36 3.6 Exercises to Section 3 .................... 38 4 Products of Schemes ......................... 40 4.1 Definition of Product . ................... 40 4.2 Group Schemes ....................... 42 4.3 Separatedness ........................ 43 4.4 Exercises to Section 4 .................... 46 6 Varieties .................................. 49 1 DefinitionsandExamples...................... 49 1.1 Definitions.......................... 49 VII VIII Contents 1.2 Vector Bundles ........................ 53 1.3 Vector Bundles and Sheaves ................. 56 1.4 Divisors and Line Bundles .................. 63 1.5 Exercises to Section 1 .................... 67 2 AbstractandQuasiprojectiveVarieties............... 68 2.1 Chow’sLemma....................... 68 2.2 BlowupAlongaSubvariety................. 70 2.3 Example of Non-quasiprojective Variety . ......... 74 2.4 CriterionsforProjectivity.................. 79 2.5 Exercises to Section 2 .................... 81 3 Coherent Sheaves .......................... 81 3.1 Sheaves of OX-Modules ................... 81 3.2 Coherent Sheaves . ................... 85 3.3 Dévissage of Coherent Sheaves ............... 88 3.4 The Finiteness Theorem ................... 92 3.5 Exercises to Section 3 .................... 93 4 Classification of Geometric Objects and Universal Schemes .... 94 4.1 Schemes and Functors . ................... 94 4.2 The Hilbert Polynomial ...................100 4.3 Flat Families .........................103 4.4 The Hilbert Scheme . ...................107 4.5 Exercises to Section 4 ....................110 Book 3: Complex Algebraic Varieties and Complex Manifolds 7 The Topology of Algebraic Varieties ..................115 1 The Complex Topology .......................115 1.1 Definitions..........................115 1.2 Algebraic Varieties as Differentiable Manifolds; Orientation..........................117 1.3 Homology of Nonsingular Projective Varieties .......118 1.4 Exercises to Section 1 ....................121 2 Connectedness ............................121 2.1 PreliminaryLemmas.....................121 2.2 The First Proof of the Main Theorem . .........122 2.3 The Second Proof . ...................124 2.4 AnalyticLemmas......................126 2.5 Connectedness of Fibres ...................127 2.6 Exercises to Section 2 ....................128 3 The Topology of Algebraic Curves .................129 3.1 Local Structure of Morphisms ................129 3.2 Triangulation of Curves ...................131 3.3 Topological Classification of Curves . .........133 3.4 Combinatorial
Load more

Recommended publications
  • MATH 145 NOTES 1. Meta Stuff and an Overview of Algebraic Geometry
    MATH 145 NOTES ARUN DEBRAY AUGUST 21, 2015 These notes were taken in Stanford’s Math 145 class in Winter 2015, taught by Ravi Vakil. I TEXed these notes up using vim, and as such there may be typos; please send questions, comments, complaints, and corrections to [email protected]. CONTENTS 1. Meta Stuff and an Overview of Algebraic Geometry: 1/5/151 2. Projective Space and Fractional Linear Transformations: 1/7/153 3. Diophantine Equations and Classifying Conics: 1/9/155 4. Quadratics and Cubics: 1/12/15 6 5. Cubics: 1/14/15 8 6. The Elliptic Curve Group and the Zariski Topology: 1/16/159 7. Affine Varieties and Noetherian Rings: 1/21/15 11 8. Localization and Varieties: 1/23/15 13 9. Hilbert’s Nullstellensatz: 1/26/15 14 10. Affine Varieties: 1/28/15 15 11. Affine Schemes: 1/30/15 17 12. Regular Functions and Regular Maps: 2/2/15 18 13. Rational Functions: 2/4/15 20 14. Rational Maps: 2/6/15 21 15. Varieties and Sheaves: 2/9/15 23 16. From Rational Functions to Rational Maps: 2/11/15 24 17. Varieties and Pre-Varieties: 2/13/15 25 18. Presheaves, Sheaves, and Affine Schemes: 2/18/15 26 19. Morphisms of Varieties and Projective Varieties: 2/20/15 28 20. Products and Projective Varieties: 2/23/15 29 21. Varieties in Action: 2/25/15 31 22. Dimension: 2/27/15 32 23. Smoothness and Dimension: 3/2/15 33 24. The Tangent and Cotangent Spaces: 3/6/15 35 25.
    [Show full text]
  • The Harvard College Mathematics Review
    The Harvard College Mathematics Review Volume 3 Spring 2011 In this issue: CHRISTOPHER POLICASTRO Artin's Conjecture ZHAO CHEN, KEVIN DONAGHUE & ALEXANDER ISAKOV A Novel Dual-Layered Approach to Geographic Profiling in Serial Crimes MICHAEL J. HOPKINS The Three-Legged Theorem MLR A Student Publication of Harvard College Website. Further information about The HCMR can be Sponsorship. Sponsoring The HCMR supports the un found online at the journal's website, dergraduate mathematics community and provides valuable high-level education to undergraduates in the field. Sponsors will be listed in the print edition of The HCMR and on a spe http://www.thehcmr.org/ fl) cial page on the The HCMR's website, (1). Sponsorship is available at the following levels: Instructions for Authors. All submissions should in clude the name(s) of the author(s), institutional affiliations (if Sponsor $0 - $99 any), and both postal and e-mail addresses at which the cor Fellow $100-$249 responding author may be reached. General questions should Friend $250 - $499 be addressed to Editor-in-Chief Rediet Abebe at hcmr@ hcs . Contributor $500-$1,999 harvard.edu. Donor $2,000 - $4,999 Patron $5,000 - $9,999 Articles. The Harvard College Mathematics Review invites Benefactor $10,000 + the submission of quality expository articles from undergrad uate students. Articles may highlight any topic in undergrad Contributors ■ Jane Street Capital • The Harvard Uni uate mathematics or in related fields, including computer sci versity Mathematics Department ence, physics, applied mathematics, statistics, and mathemat Cover Image. The image on the cover depicts several ical economics. functions, whose common zero locus describes an algebraic Authors may submit articles electronically, in .pdf, .ps, or .dvi format, to [email protected], or in hard variety (which is in this case an elliptic curve).
    [Show full text]
  • Arxiv:1507.04134V1 [Math.RA]
    NILPOTENT, ALGEBRAIC AND QUASI-REGULAR ELEMENTS IN RINGS AND ALGEBRAS NIK STOPAR Abstract. We prove that an integral Jacobson radical ring is always nil, which extends a well known result from algebras over fields to rings. As a consequence we show that if every element x of a ring R is a zero of some polynomial px with integer coefficients, such that px(1) = 1, then R is a nil ring. With these results we are able to give new characterizations of the upper nilradical of a ring and a new class of rings that satisfy the K¨othe conjecture, namely the integral rings. Key Words: π-algebraic element, nil ring, integral ring, quasi-regular element, Jacobson radical, upper nilradical 2010 Mathematics Subject Classification: 16N40, 16N20, 16U99 1. Introduction Let R be an associative ring or algebra. Every nilpotent element of R is quasi-regular and algebraic. In addition the quasi-inverse of a nilpotent element is a polynomial in this element. In the first part of this paper we will be interested in the connections between these three notions; nilpo- tency, algebraicity, and quasi-regularity. In particular we will investigate how close are algebraic elements to being nilpotent and how close are quasi- regular elements to being nilpotent. We are motivated by the following two questions: Q1. Algebraic rings and algebras are usually thought of as nice and well arXiv:1507.04134v1 [math.RA] 15 Jul 2015 behaved. For example an algebraic algebra over a field, which has no zero divisors, is a division algebra. On the other hand nil rings and algebras, which are of course algebraic, are bad and hard to deal with.
    [Show full text]
  • Solutions to Assignment 5
    Solutions to Assignment 5 Throughout, A is a commutative ring with 0 6= 1. 1. We say that a ∈ A is a zerodivisor if there exists b 6= 0 in A such that ab = 0. (This differs from Lang’s definition only to the extent that 0 will be called a zerodivisor.) Let F be the set of all ideals of A in which every element is a zerodivisor. (a) Prove that F has maximal elements. (b) Prove that every maximal element of F is a prime ideal. (c) Conclude that the set of zerodivisors is a union of prime ideals. Solution: (a) Since (0) ∈ F, the set F is nonempty. Given a chain {aα} of elements of F, it S is easily seen that α aα is an ideal as well. Since each element of this ideal is a zerodivisor, it S follows that α aα ∈ F. By Zorn’s Lemma, the set F has maximal elements. (b) Let p ∈ F be a maximal element. If x∈ / p and y∈ / p, then the ideals p + (x) and p + (y) are strictly bigger than p, and hence there exist nonzerodivisors a ∈ p + (x) and b ∈ p + (y). But then ab ∈ p + (xy) is a nonzerodivisor, and so xy∈ / p. Consequently p is a prime ideal. (c) If x ∈ A is a zerodivisor, then every element of the ideal (x) is a zerodivisor. Hence (x) ∈ F, and so (x) ⊆ p for some prime ideal p ∈ F. It follows that [ {x | x ∈ A is a zerodivisor} = p. p∈F 2. Let a be a nilpotent element of A.
    [Show full text]
  • [Nilradical of a Ring] Let N ⊂ a Consist of All Nilpotent Elements. Prove That N
    ALGEBRA 1: PROBLEM SET 10 A = a commutative ring in all the problems below. Problem 1. [Nilradical of a ring] Let n ⊂ A consist of all nilpotent elements. Prove that \ n = p p⊂A prime ideal Problem 2. Let a ⊂ A be the set of all zero{divisors of A. Is a an ideal of A? Problem 3. Let n 2 A be a nilpotent element and u 2 A be a unit (that is, u has a multiplicative inverse). Prove that u + n is again a unit. Problem 4. Let S ⊂ A be a set such that 0 62 S and r; s 2 S implies rs 2 S (multiplicatively closed set). Let p be an ideal which is maximal among the ideals not intersecting S. That is, maximal with respect to inclusion, from the following set: IS = fa ⊂ A an ideal such that a \ S = ;g Prove that p is prime. Problem 5. If for every x 2 A there exists an n ≥ 2 such that xn = x, then prove that every prime ideal in A is maximal. Problem 6. Let B be another commutative ring and let f : A ! B be a ring homomorphism. Let p ⊂ A be a prime ideal and define b ⊂ B to be the ideal generated by f(p). Prove or disprove: b is a prime ideal. Problem 7. Let K be a field and R = K[[x]] be the ring of formal power series in a variable x with coefficients from K. Prove that R is a local ring, with unique maximal ideal m = (x).
    [Show full text]
  • Gps Abh1.Pdf
    1 Lecture L6: Pause and Refresh By reading the following summaries of the first five lectures, the rest of the book may become intelligible without studying the details of those lectures. In the first lecture we have introduced the basic structures of algebra such as groups, rings, fields, vector spaces, ideals, modules, polynomials, rational functions, euclidean domains, principal ideal domains, and unique factorization domains. In the second lecture, after introducing power series, meromorphic series, and valuations, we show the equivalence of well-ordering and Zorn's Lemma and use them to establish the existence of vector space basis, transcendence basis, algebraic closure, and maximal ideals. The third lecture deals with the power series theorems of Newton, Hensel, and Weierstrass. The fourth and fifth lectures deal with ideals, modules, varieties, and models which are the avatars of varieties full of local rings. x1: SUMMARY OF LECTURE L1 ON QUADRATIC EQUATIONS For sets (= collections of objects) S and T , a map φ : S ! T is an assignment which to every x 2 S, i.e., to every element (= object) x of S, assigns φ(x) 2 T ; this may be written x 7! φ(x); the element φ(x) is called the image of x under φ; we put dom(φ) = S and ran(φ) = T and call these the domain and range of φ respectively. The composition of maps φ : S ! T and : T ! U is the map φ : S ! U given by ( φ)(x) = (φ(x)). The map φ is injective, or is an injection, if φ(x) = φ(y) ) x = y.A subset of S is a set R whose objects are amongst the objects of S; we write R ⊂ S; we may also write S ⊃ R and call S an overset of R.
    [Show full text]
  • A Primer of Commutative Algebra
    A Primer of Commutative Algebra James S. Milne August 15, 2014, v4.01 Abstract These notes collect the basic results in commutative algebra used in the rest of my notes and books. Although most of the material is standard, the notes include a few results, for example, the affine version of Zariski’s main theorem, that are difficult to find in books. Contents 1 Rings and algebras......................3 2 Ideals...........................3 3 Noetherian rings.......................9 4 Unique factorization...................... 14 5 Rings of fractions....................... 18 6 Integral dependence...................... 24 7 The going-up and going-down theorems.............. 30 8 Noether’s normalization theorem................. 33 9 Direct and inverse limits.................... 35 10 Tensor Products....................... 38 11 Flatness.......................... 43 12 Finitely generated projective modules............... 49 13 Zariski’s lemma and the Hilbert Nullstellensatz............ 57 14 The spectrum of a ring..................... 60 15 Jacobson rings and max spectra.................. 68 16 Artinian rings........................ 73 17 Quasi-finite algebras and Zariski’s main theorem............ 74 18 Dimension theory for finitely generated k-algebras........... 81 19 Primary decompositions.................... 86 20 Dedekind domains...................... 91 21 Dimension theory for noetherian rings............... 96 22 Regular local rings...................... 100 23 Flatness and fibres...................... 102 24 Completions......................... 105 A Solutions to the exercises..................... 107 References........................... 108 Index............................. 109 c 2009, 2010, 2011, 2012, 2013, 2014 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permission from the copyright holder. Available at www.jmilne.org/math/. 1 CONTENTS 2 Notations and conventions Our convention is that rings have identity elements,1 and homomorphisms of rings respect the identity elements.
    [Show full text]
  • Determining Corresponding Artinian Rings to Zero-Divisor Graphs Abigail Mary Jones Wellesley College, [email protected]
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Wellesley College Wellesley College Wellesley College Digital Scholarship and Archive Honors Thesis Collection 2016 Determining Corresponding Artinian Rings to Zero-Divisor Graphs Abigail Mary Jones Wellesley College, [email protected] Follow this and additional works at: https://repository.wellesley.edu/thesiscollection Recommended Citation Jones, Abigail Mary, "Determining Corresponding Artinian Rings to Zero-Divisor Graphs" (2016). Honors Thesis Collection. 341. https://repository.wellesley.edu/thesiscollection/341 This Dissertation/Thesis is brought to you for free and open access by Wellesley College Digital Scholarship and Archive. It has been accepted for inclusion in Honors Thesis Collection by an authorized administrator of Wellesley College Digital Scholarship and Archive. For more information, please contact [email protected]. Determining Corresponding Artinian Rings to Zero-Divisor Graphs Author: Abigail Mary Jones Advisor: Professor Alexander Diesl April 2016 Submitted in Partial Fulfillment of the Prerequisite for Honors in the Department of Mathematics Wellesley College c 2016 Abigail Jones Acknowledgements I would like to thank my advisor, Professor Diesl, who helped me throughout the process, guided me on my research, read my drafts, and fueled my enthusiasm for zero-divisor graphs. I would also like to thank my committee members Professor Phyllis McGibbon, Professor Megan Kerr, and Professor Andrew Schultz. Also, a very special thank you to my friends, who shaped my Wellesley experience and supported me for the past four years, and to my family, for their unwavering love and support. Abstract We introduce Anderson's and Livingston's definition of a zero-divisor graph of a commutative ring.
    [Show full text]
  • 1. Introduction This Paper Given a Ring R We Use Represent the Prime
    East Asian Math J 18(2002), No 1, pp. 15-20 RINGS IN WHICH NILPOTENT ELEMENTS FORM AN IDEAL Jung Rae Cho, Nam Kyun Kim and Yang Lee Abstract We study the relation아 ups between strongly prime ideals and completely prime ideals, concentrating on the connections among various radicals (prime radical, upper nilradical and generalized nilrad- ical). Given a ring R, consider the condition (*) nilpotent elements of R form an ideal m R. We show that a ring R satisfies (*) if and only if^every mmimaLsfrmgly 一prime一 丄deal of R is completely prime_ifLand only if the upper nilradical coincides with the generalized nilradical in R 1. Introduction This paper was motivated by the results in [1] and [4] which are related to nilradicals. Throughout this paper, all rings are associative with identity. Given a ring R we use P(B), N(B), and Specs(R) to represent the prime radical, the set of all nilpotent elements, and 난 set of all strongly prime ideals of R)respectively. Recall that a ring R is called strongly prime if R is prime with no nonzero nil ideals and an ideal F of 7? is called strongly prime if R/P is strongly prime, and that the upper nilradical of a ring R is the unique Received November 12, 2001 2000 Mathematics Subject Classification: 16D25, 16N40, 16N60. Key words and phrases, upper nilradical, strongly prime ideal, completely prime ideal. The first and third named authors were supported by Korea Research Foun­ dation Grant (KRF-2000-015-DP0002), and the third named author was also sup­ ported by grant No.(R02-2000-00014) from the Korea Science & Engineering Foundation.
    [Show full text]
  • Some Results on the Complement of a New Graph Associated to a Commutative Ring
    Communications in Combinatorics and Optimization Vol. 2 No. 2, 2017 pp.119-138 CCO DOI: 10.22049/CCO.2017.25908.1053 Commun. Comb. Optim. Some results on the complement of a new graph associated to a commutative ring S. Visweswaran1∗ and A. Parmar1 1 Department of Mathematics, Saurashtra University, Rajkot, India, 360 005 s [email protected] Received: 7 March 2017; Accepted: 29 July 2017; Published Online: 3 August 2017 Communicated by Seyed Mahmoud Sheikholeslami Abstract: The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal. Let R be a ring. We denote the collection of ∗ all ideals of R by I(R) and I(R)nf(0)g by I(R) . Alilou et al. [A. Alilou, J. Amjadi and S.M. Sheikholeslami, A new graph associated to a commutative ring, Discrete Math. Algorithm. Appl. 8 (2016) Article ID: 1650029 (13 pages)] introduced and investigated ∗ a new graph associated to R, denoted by ΩR which is an undirected graph whose vertex ∗ set is I(R) nfRg and distinct vertices I;J are joined by an edge in this graph if and only if either (AnnRI)J = (0) or (AnnRJ)I = (0). Several interesting theorems were ∗ proved on ΩR in the aforementioned paper and they illustrate the interplay between ∗ the graph-theoretic properties of ΩR and the ring-theoretic properties of R. The aim ∗ c of this article is to investigate some properties of (ΩR) , the complement of the new ∗ graph ΩR associated to R. Keywords: Annihilating ideal of a ring, maximal N-prime of (0), special principal ideal ring, connected graph, diameter, girth AMS Subject classification: 13A15, 05C25 1.
    [Show full text]
  • Solutions to Assignment 5
    Solutions to Assignment 5 Throughout, A is a commutative ring with 0 6= 1. 1. We say that a ∈ A is a zerodivisor if there exists b 6= 0 in A such that ab = 0. (This differs from Lang’s definition only to the extent that 0 will be called a zerodivisor.) Let F be the set of all ideals of A in which every element is a zerodivisor. (a) Prove that F has maximal elements. (b) Prove that every maximal element of F is a prime ideal. (c) Conclude that the set of zerodivisors is a union of prime ideals. Solution: (a) Since (0) ∈ F, the set F is nonempty. Given a chain {aα} of elements S of F, it is easily seen that aα is an ideal as well. Since each element of this ideal α S is a zerodivisor, it follows that α aα ∈ F. By Zorn’s Lemma, the set F has maximal elements. (b) Let p ∈ F be a maximal element. If x∈ / p and y∈ / p, then the ideals p + (x) and p+(y) are strictly bigger than p, and hence there exist nonzerodivisors a ∈ p+(x) and b ∈ p + (y). But then ab ∈ p + (xy) is a nonzerodivisor, and so xy∈ / p. Consequently p is a prime ideal. (c) If x ∈ A is a zerodivisor, then every element of the ideal (x) is a zerodivisor. Hence (x) ∈ F, and so (x) ⊆ p for some prime ideal p ∈ F. It follows that [ {x | x ∈ A is a zerodivisor} = p. p∈F 2. Let a be an ideal of A.
    [Show full text]
  • On the Jacobson Radical of Skew Polynomial Extensions of Rings Satisfying a Polynomial Identity
    ON THE JACOBSON RADICAL OF SKEW POLYNOMIAL EXTENSIONS OF RINGS SATISFYING A POLYNOMIAL IDENTITY BLAKE W. MADILL Abstract. Let R be a ring satisfying a polynomial identity and let D be a derivation of R. We consider the Jacobson radical of the skew polynomial ring R[x; D] with coefficients in R and with respect to D, and show that J(R[x; D])\ R is a nil D-ideal. This extends a result of Ferrero, Kishimoto, and Motose, who proved this in the case when R is commutative. 1. Introduction Let R be a ring (not necessarily unital). We say that a map D : R ! R is a derivation if it satisfies D(a + b) = D(a) + D(b) and D(ab) = D(a)b + aD(b) for all a; b 2 R. Given a derivation D of a ring R, we can construct the differential polynomial ring, also known as a skew polynomial ring of derivation type, R[x; D], which as a set is given by n fx an + ··· + xa1 + a0 : n ≥ 0; ai 2 Rg; with regular polynomial addition and multiplication given by xa = ax + D(a) for a 2 R and then extending via associativity and linearity. When D is the trivial derivation of R (i.e. D(a) = 0 for all a 2 R) then R[x; D] is simply R[x]. In this case, Amitsur [1] showed that the Jacobson radical J(R[x]) is the polynomial ring (J(R[x]) \ R)[x] and that J(R[x]) \ R is a nil ideal of R.
    [Show full text]