DOCSLIB.ORG

Rings and Their Modules

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Rings and Their Modules De Gruyter Graduate Paul E. Bland Rings and Their Modules De Gruyter Mathematics Subject Classification 2010: Primary: 16-01; Secondary: 16D10, 16D40, 16D50, 16D60, 16D70, 16E05, 16E10, 16E30. ISBN 978-3-11-025022-0 e-ISBN 978-3-11-025023-7 Library of Congress Cataloging-in-Publication Data Bland, Paul E. Rings and their modules / by Paul E. Bland. p. cm. Ϫ (De Gruyter textbook) Includes bibliographical references and index. ISBN 978-3-11-025022-0 (alk. paper) 1. Rings (Algebra) 2. Modules (Algebra) I. Title. QA247.B545 2011 5121.4Ϫdc22 2010034731 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.d-nb.de. ” 2011 Walter de Gruyter GmbH & Co. KG, Berlin/New York Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen ϱ Printed on acid-free paper Printed in Germany www.degruyter.com Preface The goal of this text is to provide an introduction to the theory of rings and modules that goes beyond what one normally obtains in a beginning graduate course in abstract algebra. The author believes that a text directed to a study of rings and modules would be deficient without at least an introduction to homological algebra. Such an introduction has been included and topics are intermingled throughout the text that support this introduction. An effort has been made to write a text that can, for the most part, be read without consulting references. No attempt has been made to present a survey of rings and/or modules, so many worthy topics have been omitted in order to hold the text to a reasonable length. The theme of the text is the interplay between rings and modules. At times we will investigate a ring by considering a given set of conditions on the modules it admits and at other times we will consider a ring of a certain type to see what structure is forced on its modules. About the Text It is assumed that the reader is familiar with concepts such as Zorn’s lemma, commu- tative diagrams and ordinal and cardinal numbers. A brief review of these and other basic ideas that will hold throughout the text is given in the chapter on preliminaries to the text and in Appendix A. We also assume that the reader has a basic knowledge of rings and their homomorphisms. No such assumption has been made with regard to modules. In the first three sections of Chapter 1, the basics of ring theory have been provided for the sake of completeness and in order to give readers quick access to topics in ring theory that they might require to refresh their memory. An introduction to the fundamental properties of (unitary) modules, submodules and module homomorph- isms is also provided in this chapter. Chapters 1 through 6 present what the author considers to be a “standard” development of topics in ring and module theory, culmi- nating with the Wedderburn–Artin structure theorems in Chapter 6. These theorems, some of the most beautiful in all of abstract algebra, present the theory of semisimple rings. Over such a ring, modules exhibit properties similar to those of vector spaces: submodules are direct summands and every module decomposes as a direct sum of simple submodules. Topics are interspersed throughout the first six chapters that sup- port the development of semisimple rings and the accompanying Wedderburn-Artin theory. For example, concepts such as direct products, direct sums, free modules and tensor products appear in Chapter 2. vi Preface Another goal of the text is to give a brief introduction to category theory. For the most part, only those topics necessary to discuss module categories are developed. However, enough attention is devoted to categories so that the reader will have at least a passing knowledge of this subject. Topics in category theory form the substance of Chapter 3. Central to any study of semisimple rings are ascending and descending chain con- ditions on rings and modules, and injective and projective modules. These topics as well as the concept of a flat module are covered in Chapters 4 and 5. Our investigation of semisimple rings begins in Chapter 6 with the development of the Jacobson radical of a ring and the analogous concept for a module. Simple artinian rings and primitive rings are also studied here and it is shown that a ring is semisimple if and only if it is a finite ring direct product of n n matrix rings each with entries from a division ring. The remainder of the text is a presentation of various topics in ring and module the- ory, including an introduction to homological algebra. These topics are often related to concepts developed in Chapters 1 through 6. Chapter 7 introduces injective envelopes and projective covers. Here it is shown that every module has a “best approximation” by an injective module and that, for a particular type of ring, every module has a “best approximation” by a projective module. Quasi-injective and quasi-projective covers are also developed and it is es- tablished that every module has a projective cover if and only if every module has a quasi-projective cover. In Chapter 8 a localization procedure is developed that will produce a ring of frac- tions (or quotients) of a suitable ring. This construction, which is a generalization of the method used to construct the field of fractions of an integral domain, plays a role in the study of commutative algebra and, in particular, in algebraic geometry. Chapter 9 is devoted to an introduction to graded rings and modules. Graded rings and modules are important in the study of commutative algebra and in algebraic ge- ometry where they are used to gain information about projective varieties. Many of the topics introduced in Chapters 1 through 8 are reformulated and studied in this “new” setting. Chapter 10 deals with reflexive modules. The fact that a vector space is reflexive if and only it is finite dimensional, leads naturally to the question, “What are the rings over which every finitely generated module is reflexive?” To this end, quasi- Frobenius rings are defined and it is proved that if a ring is quasi-Frobenius, then every finitely generated module is reflexive. However, the converse fails. Consideration of a converse is taken up in Chapter 12 after our introduction to homological algebra has been completed and techniques from homological algebra are available. The substance of Chapter 11 is homological algebra. Projective and injective reso- lutions of modules are established and these concepts are used to investigate the left and right derived functors of an additive (covariant/contravariant) functor F .These Preface vii n results are then applied to obtain ExtR,thenth right derived functors of Hom, and R Torn ,thenth left derived functors of ˝R. The text concludes with Chapter 12 where an injective, a projective and a flat di- mension of a module are defined. A right global homological dimension of a ring is also developed as well as a global flat (or weak) dimension of a ring. It is shown that these dimensions can be used to gain information about some of the rings and modules studied in earlier chapters. In particular, using homological methods it is shown that if a ring is left and right noetherian and if every finitely generated module is reflexive, then the ring is quasi-Frobenius. There are several excellent texts given in the bibliography that can be consulted by the reader who wishes additional information on rings and modules and homological algebra. Problem Sets Problem sets follow each section of the text and new topics are sometimes introduced in the problem sets. These new topics are related to the material given in the section and they are often an extension of that material. For some of the exercises, a hint as to how one might begin to write a solution for the exercise is given in brackets at the end of the exercise. Often such a hint will point to a result in the text that can be used to solve the exercise, or the hint will point to a result in the text whose proof will suggest a technique for writing a solution. It is left to the reader to de- cide which is the case. Finally, the exercises presented in the problem sets for Sec- tions 1.1, 1.2 and 1.3 are intended as a review. The reader may select exercises from these problem sets according to their interests or according to what they feel might be necessary to refresh their memories on the arithmetic of rings and their homomorph- isms. Cross Referencing The chapters and sections of the text have been numbered consecutively while propo- sitions and corollaries have been numbered consecutively within each section. Ex- amples have also been numbered consecutively within each section and referenced by example number and section number of each chapter unless it is an example in the current section. For instance, Example 3 means the third example in the current section while Example 3 in Section 4.1 has the obvious meaning. Similar remarks hold for the exercises in the problem sets. Also, some equations have been numbered on their right. In each case, the equations are numbered consecutively within each chapter without regard for the chapter sections. viii Preface Backmatter The backmatter for the text is composed of a bibliography, a section on ordinal and cardinal numbers, a list of symbols used in the text, and a subject matter index.
Load more

Recommended publications
  • 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].
    [Show full text]
  • Lectures on Non-Commutative Rings
    Lectures on Non-Commutative Rings by Frank W. Anderson Mathematics 681 University of Oregon Fall, 2002 This material is free. However, we retain the copyright. You may not charge to redistribute this material, in whole or part, without written permission from the author. Preface. This document is a somewhat extended record of the material covered in the Fall 2002 seminar Math 681 on non-commutative ring theory. This does not include material from the informal discussion of the representation theory of algebras that we had during the last couple of lectures. On the other hand this does include expanded versions of some items that were not covered explicitly in the lectures. The latter mostly deals with material that is prerequisite for the later topics and may very well have been covered in earlier courses. For the most part this is simply a cleaned up version of the notes that were prepared for the class during the term. In this we have attempted to correct all of the many mathematical errors, typos, and sloppy writing that we could nd or that have been pointed out to us. Experience has convinced us, though, that we have almost certainly not come close to catching all of the goofs. So we welcome any feedback from the readers on how this can be cleaned up even more. One aspect of these notes that you should understand is that a lot of the substantive material, particularly some of the technical stu, will be presented as exercises. Thus, to get the most from this you should probably read the statements of the exercises and at least think through what they are trying to address.
    [Show full text]
  • ON the LEVITZKI RADICAL of MODULES Nico J. Groenewald And
    International Electronic Journal of Algebra Volume 15 (2014) 77-89 ON THE LEVITZKI RADICAL OF MODULES Nico J. Groenewald and David Ssevviiri Received: 5 May 2013; Revised: 15 August 2013 Communicated by Abdullah Harmancı Dedicated to the memory of Professor Efraim P. Armendariz Abstract. In [1] a Levitzki module which we here call an l-prime module was introduced. In this paper we define and characterize l-prime submodules. Let N be a submodule of an R-module M. If p l: N := fm 2 M : every l- system of M containingm meets Ng; p we show that l: N coincides with the intersection L(N) of all l-prime submod- ules of M containing N. We define the Levitzki radical of an R-module M as p L(M) = l: 0. Let β(M), U(M) and Rad(M) be the prime radical, upper nil radical and Jacobson radical of M respectively. In general β(M) ⊆ L(M) ⊆ U(M) ⊆ Rad(M). If R is commutative, β(M) = L(M) = U(M) and if R is left Artinian, β(M) = L(M) = U(M) = Rad(M). Lastly, we show that the class of all l-prime modules RM with RM 6= 0 forms a special class of modules. Mathematics Subject Classification (2010): 16D60, 16N40, 16N60, 16N80, 16S90 Keywords: l-prime submodule; semi l-prime; s-prime submodule; upper nil radical; Levitzki radical 1. Introduction All modules are left modules, the rings are associative but not necessarily unital. By I/R and N ≤ M we respectively mean I is an ideal of a ring R and N is a submodule of a module M.
    [Show full text]
  • Noncomrnutative Algebra
    Graduate Texts in Mathematics 144 Editorial Board 1. H. Ewing F. W. Gehring P. R. Halmos Graduate Texts in Mathematics TAKEUTI/ZARING. Inlroduction to Axiomatic Set Theory. 2nd ed. 2 OXTOIlY. Measure and Category. 2nd ed. 3 SCHAEFfER. Topological Vector Spaces. 4 HILTON/STAMMBACH. A Course in Homological Algebra. 5 MAC LANE. Categories for the Working Mathematician. 6 HUGHES/PIPER. Projective Planes. 7 SERRE. A Course in Arithmetic. 8 TAKEUTt/ZARING. Axiometic Set Theory. 9 HUMPHREYS. Introduction to Lie Algebras and Representation Theory. 10 COHEN. A Course in Simple Homotopy Theory. 11 CONWAY. Functions of One Complex Variable. 2nd ed. 12 BEALS. Advanced Mathematical Analysis. 13 ANDERSON/FuLLER. Rings and Categories of Modules. 2nd ed. 14 GOLUBITSKy/GUILEMIN. Stable Mappings and Their Singularities. 15 BERBERIAN. Lectures in Functional Analysis and Operator Theory. 16 WINTER. The Structure of Fields. 17 ROSENIlLATT. Random Processes. 2nd ed. 18 HALMos. Measure Theory. 19 HALMOS. A Hilbert Space Problem Book. 2nd ed. 20 HUSEMOLLER. Fibre Bundles. 2nd ed. 21 HUMPHREYS. Linear Algebraic Groups. 22 BARNES/MACK. An Algebraic Introduction to Mathematical Logic. 23 GREUB. Linear Algebra. 4th ed. 24 HOLMES. Geometric Functional Analysis and Its Applications. 25 HEWITTISTROMBERG. Real and Abstract Analysis. 26 MANES. Algebraic Theories. 27 KELLEY. General Topology. 28 ZARISKI/SAMUEL. Commutative Algebra. Vol. I. 29 ZARISKI/SAMUEL. Commutative Algebra. Vol. II. 30 JACOIlSON. Lectures in Abstract Algebra I. Basic Concepts. 31 JACOBSON. Lectures in Abstract Algebra II. Linear Algebra. 32 JACOBSON. Lectures in Abstract Algebra III. Theory of Fields and Galois Theory. 33 HIRSCH. Differential Topology. 34 SPITZER. Principles of Random Walk.
    [Show full text]
  • Noncommutative Algebra 2: Representations of Finite-Dimensional Algebras
    Noncommutative Algebra 2: Representations of finite-dimensional algebras Bielefeld University, Summer Semester 2017 William Crawley-Boevey 1 Introduction Let K be a field. Often it will be algebraically closed. Let A be a K- algebra. Any A-module M becomes a vector space. We are interested in the classification of finite-dimensional A-modules, especially in the case when A is finite-dimensional. References: D. J. Benson, Representations and cohomology Assem, Simson and Skowronski, Elements of the representation theory of associative algebras I Auslander, Reiten and Smalo, Representation theory of artin algebras R. Schiffler, Quiver representations 1.1 Examples of algebras (1) The group algebra A = KG for a group G. Then A-modules are the same as representations of the group G, so group homomorphisms G ! GL(V ). (2) Let V = Kn and let q : V ! K be a quadratic form. The associated Clifford algebra CK (V; q) is T V=I, where TV is the tensor algebra on V over K and I is the ideal generated by v2 − q(v)1 for all v 2 V . For example if Pn Pn 2 V has basis e1; : : : ; en and q( i=1 λiei) = i=1 aiλi for all λ1; : : : ; λn 2 K, 2 then C(V; q) is generated by e1; : : : ; en subject to the relations ei = ai1 1 and eiej = −ejei for i 6= j. It has basis the products ei1 ei2 : : : eik with i1 < i2 < ··· < ik. In particular the exterior algebra is the case when q = 0. 2 2 2 For example CR(R ; q) with q(x1; x2) = −x1 − x2 is the quaternions.
    [Show full text]
  • Semisimplicity and Some Basic Structure Theorems
    CHAPTER 2 Semisimplicity and some basic structure theorems In this chapter we will review some of the classical structure theorems for finite dimensional algebras. In most cases results will be stated with only a sketch of the proof. Henceforth we will restrict our attention to finite dimensional modules. 2.1. Simple modules and semisimplicity Recall that a simple module is a module S such that the only submodules are S and 0. These form the building blocks out of which all other modules are made: LEMMA 2.1.1. If M is a finite dimensional A-module then there exists a sequence of submod- ules 0 = M0 ⊂ M1 ⊂···⊂ Mn = M such that Mi/Mi−1 is simple for each 1 ≤ i ≤ n. Such a series is called a composition series for M. PROOF. Proceed by induction on the dimension of M. If M is not simple, pick a submodule M1 of minimal dimension, which is necessarily simple. Now dim(M/M1) < dimM, and so the result follows by induction. Moreover, we have THEOREM 2.1.2 (Jordan-Hölder). Suppose that M has two composition series 0 = M0 ⊂ M1 ⊂···⊂ Mm = M, 0 = N0 ⊂ N1 ⊂···⊂ Nn = M. Then n = m and there exists a permutation σ of {1,...n} such that ∼ Mi/Mi+1 = Nσ(i)/Nσ(i)+1. PROOF. The proof is similar to that for groups. Life would be (relatively) straightforward if every module was a direct sum of simple modules. DEFINITION 2.1.3. A moduleM is semisimple (or completely reducible)if it can be written as a direct sum of simple modules.
    [Show full text]
  • Noted Ord A, Is the Minimum N > 0 Such That an = 1
    210ABC (17F/18W/18S) (L) A. Merkurjev / (TA) A. Wertheim Algebra Contents 1 Groups 3 1.1 Definitions and basic properties..............................3 1.2 Examples of groups.....................................4 1.3 Homomorphisms......................................5 1.4 Cyclic groups........................................6 1.5 Subgroups..........................................6 1.6 Cosets and Lagrange's theorem..............................8 1.7 Normal subgroups.....................................9 1.8 Isomorphism theorems................................... 10 1.9 Group actions........................................ 12 1.10 p-groups and the Sylow theorems............................. 15 1.11 Direct products....................................... 17 1.12 Nilpotent and solvable groups............................... 20 1.13 Symmetric and alternating groups............................ 22 1.14 Semidirect products.................................... 25 1.15 Groups of small order................................... 27 1.16 Exact sequences....................................... 29 1.17 Free groups......................................... 30 2 Categories and Functors 35 2.1 Definitions and basic properties.............................. 35 2.2 Products and coproducts.................................. 37 2.3 Functors........................................... 40 2.4 Natural transformations.................................. 41 2.5 Limits............................................ 44 3 Rings 47 3.1 Definitions and basic properties.............................
    [Show full text]
  • 8 the Socle and Radical
    8 The Socle and Radical. Simple and semisimple modules are clearly the main building blocks in much of ring theory. Of coure, not every module can be built from semisimple modules, but for many modules its semisimple submodules and semisimple factor modules play important roles in understanding the module. Here we look briey at two submodules of a module M that provide us with important preliminary information about the extendt to which semisimple modules determine its structure. Let RM be a module. Then its socle is the submodule X Soc M = {N N : N is simple}. So the socle of M is the largest submodule of M generated by simple modules, or equivalently, it is the largest semisimple submodule of M. Dually, the radical of M is the submodule \ Rad M = {N M : N is maximal in M}. So the radical of M is the smallest submodule of M modulo which M is cogenerated by simples, or equivalently, it is the smallest submodule modulo which M can be embedded in a product of simples. 8.1. Lemma. Let f : M → N be an R-homomorphism. Then f(Soc M) Soc N and f(Rad M) Rad N. Thus, if RMS is a bimodule, then both Soc(RM) and Rad(RM) are (R, S) sub bimodules of M. Proof. The rst assertion is a trivial consequence of Schur’s Lemma (Lemma 7.1). For the second, observe that for every simple RT and every R-homomorphism g : N → T , g f : M → T ,sowehave Rad M Ker(g f), and f(Rad M) Ker g.
    [Show full text]
  • Rings Characterized by Their Modules
    Rings Characterized by Their Modules A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Christopher J. Holston August 2011 © 2011 Christopher J. Holston. All Rights Reserved. 2 This dissertation titled Rings Characterized by Their Modules by CHRISTOPHER J. HOLSTON has been approved for the Department of Mathematics and the College of Arts and Sciences by Dinh van Huynh Professor of Mathematics Sergio Lopez-Permouth´ Professor of Mathematics Benjamin M. Ogles Dean for College of Arts and Sciences 3 Abstract HOLSTON, CHRISTOPHER J., Ph.D., August 2011, Mathematics Rings Characterized by Their Modules (48 pp.) Directors of Dissertation: Dinh van Huynh and Sergio Lopez-Permouth´ A ring R is called a right WV-ring if each simple right R-module is injective relative to proper cyclics. A right WV-ring which is not a right V-ring is called a right strictly WV-ring. A right strictly WV-ring R has only three two-sided ideals. If, in addition, R J(R) is finitely generated, then it has only three right ideals. It is shown that, given a cyclic right module C over a right WV-ring R, C is noetherian iff every cyclic module in σ[C] is a direct sum of a module which is projective in σ[C] with a module which is either CS or has finite uniform dimension. A module is called p-poor if it is projective only with respect to the semisimple modules. Every ring has a semisimple p-poor module.
    [Show full text]
  • 10. the Radical Let a Be a Ring (With Identity According to Our Convention
    10. The Radical (Apres` Bourbaki Algebra Chapter 8 §6) [s:rad] Let A be a ring (with identity according to our convention) and M an A-module. For additive subgroups U and V of A and M respectively, we denote by UV the P subset of M consisting of finite sums i uivi with ui and vi in U and V respectively. Thus UV is a submodule if U is a left ideal; the product of left ideals is a left ideal; the product of a left ideal and a right ideal is a two-sided ideal. [ss:nil] 10.1. Nil and nilpotent ideals. An element a (respectively, an ideal a) of A is nilpotent if an = 0 (respectively, an = 0) for some n ≥ 1. An ideal consisting of nilpotent elements is a nil ideal. We have: • If a is nilpotent, then 1 − a is a unit with inverse 1 + a1 + a2 + ... (note that the sum is finite). • An ideal is nilpotent if and only if there exists an n such that the product of any n elements all belonging to the ideal vanishes. • A nilpotent ideal is clearly nil. But not every nil ideal is nilpotent. In fact, in §?? we exhibit a non-zero nil ideal a such that a2 = a. [ss:radm] 10.2. The radical of a module. The radical Rad M of a module M is the inter- section of all its maximal submodules, or, equivalently, the intersection of kernels of all homomorphisms into simple modules. We have: • Rad M vanishes if and only if M is the submodule of a direct product of simple modules; in particular, a semisimple module has trivial radical.
    [Show full text]
  • Chapter One Basic Definitions and Results
    Republic of Iraq Ministry of Higher Education & Scientific Research University of Al-Qadisiyah College of Computer Science and Mathematics Department of Mathematics SS-Injective Modules and Related Concepts A thesis Submitted to the Council of the College of Computer Science and Mathematics, University of Al-Qadisiyah as partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics By Adel Salim Tayyah Supervised by Assist. Prof. Dr. Akeel Ramadan Mehdi October 2016 A.D. ﴿يَزْفَعِ اللَّهُ الَّذِيهَ آَمَنُوا مِنْكُمْ وَالَّذِيهَ أُوتُوا الْعِلْمَ دَرَجَاتٍ وَاللَّهُ بِمَا تَعْمَلُونَ خَبِريٌ﴾ ) اجملادلة :۱۱( Acknowledgements Praise should be first to Almighty Allah for all the strength, patience and insistence He bestowed on me, which enabled me to complete the present study. Thanks to him until He is pleased. Heavy debt of gratitude and profound thankfulness must go to my supervisor Assist. Prof. Dr. Akeel Ramadan Mehdi for his professionalism, fruitful suggestions, invaluable and knowledgeable remarks, precious advice, insightful supervision, constant encouragement and accurate revisions, which have contributed a lot to the development of the present work and paved the way for its completion. My special thanks must also go to my wife for providing the academic support, and also for sharing me the good and bad times during the preparation of this study. I am wholeheartedly grateful to whoever contributed academically, directly or indirectly, in producing the final form of the thesis. In view of the available recommendation, I forward this thesis for debate by the examining committee. Head of the Mathematics Department Signature: Name: Dr. Qusuay Hatim Eqaar Date: / / 2016 Dept.
    [Show full text]