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Classes of Modules Copyright 2006 by Taylor & Francis Group, LLC PURE AND APPLIED MATHEMATICS A Program of Monographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS Earl J. Taft Zuhair Nashed Rutgers University University of Central Florida Piscataway, New Jersey Orlando, Florida EDITORIAL BOARD M. S. Baouendi Freddy van Oystaeyen University of California, University of Antwerp, San Diego Belgium Jane Cronin Donald Passman Rutgers University University of Wisconsin, Madison Jack K. Hale Georgia Institute of Technology Fred S. Roberts Rutgers University S. Kobayashi University of California, David L. Russell Berkeley Virginia Polytechnic Institute and State University Marvin Marcus University of California, Walter Schempp Santa Barbara Universität Siegen W. S. Massey Mark Teply Yale University University of Wisconsin, Milwaukee Anil Nerode Cornell University Copyright 2006 by Taylor & Francis Group, LLC MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS Recent Titles G. S. 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Hastings, Introduction to the Mathematics of Operations Research with Mathematica®, Second Edition (2006) Robert Carlson, A Concrete Introduction to Real Analysis (2006) John Dauns and Yiqiang Zhou, Classes of Modules (2006) Copyright 2006 by Taylor & Francis Group, LLC Classes of Modules John Dauns Tulane University New Orleans, Louisiana, U.S.A. Yiqiang Zhou Memorial University of Newfoundland St. John’s, Canada Boca Raton London New York Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business Copyright 2006 by Taylor & Francis Group, LLC Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor and Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 1-58488-660-9 (Hardcover) International Standard Book Number-13: 978-1-58488-660-0 (Hardcover) Library of Congress Card Number 2006045438 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the conse- quences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Dauns, John. Classes of modules / John Dauns, Yiqiang Zhou. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-58488-660-0 (acid-free paper) ISBN-10: 1-58488-660-9 (acid-free paper) 1. Set theory. 2. Modules (Algebra) 3. Rings (Algebra) I. Zhou, Yiqiang. II. Title. QA248.D275 2006 512’.42--dc22 2006045438 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Copyright 2006 by Taylor & Francis Group, LLC C6609_Discl.indd 1 5/16/06 3:07:07 PM Contents Preface v Note to the Reader vii List of Symbols ix Chapter 1. PreliminaryBackground 1 1.1. Notation and Terminology 1 1.2. Lattices 5 Chapter 2. ImportantModuleClassesandConstructions 7 2.1. Torsion Theory 7 2.2. Module Class σ[M] 12 2.3. Natural Classes 19 2.4. M-Natural Classes 24 2.5. Pre-Natural Classes 29 Chapter 3. FinitenessConditions 33 3.1. Ascending Chain Conditions 33 3.2. Descending Chain Conditions 50 3.3. Covers and Ascending Chain Conditions 65 Chapter 4. TypeTheoryofModules: Dimension 71 4.1. Type Submodules and Type Dimensions 71 4.2. Several Type Dimension Formulas 80 4.3. Some Non-Classical Finiteness Conditions 87 iii Copyright 2006 by Taylor & Francis Group, LLC iv Chapter 5. TypeTheoryofModules: Decompositions 107 5.1. Type Direct Sum Decompositions 108 5.2. Decomposability of Modules 117 5.3. Unique Type Closure Modules 132 5.4. TS-Modules 142 Chapter 6. LatticesofModuleClasses 149 6.1. Lattice of Pre-Natural Classes 149 6.2. More Sublattice Structures 153 6.3. Lattice Properties of Np(R) 163 r 6.4. More Lattice Properties of Np(R) 177 r 6.5. Lattice N (R) and Its Applications 190 r 6.6. Boolean Ideal Lattice 200 References 205 Copyright 2006 by Taylor & Francis Group, LLC Preface The main theme of the book is in two concepts and how they pervade and structuremuchofringandmoduletheory. Theyareanaturalclass,andatype submodule. A natural class K of right modules over an arbitrary associative ringRwithidentityisonethatisclosedunderisomorphiccopies,submodules, arbitrarydirectsums,andinjectivehulls. AsubmoduleN ofarightR-module M is a type submodule if there exists some natural class K such that N ⊆M is a submodule maximal with respect to the property that N ∈K. There are also equivalent but somewhat technical ways of defining this notion totally internally in terms of the given module M without reference to any outside classes K. Equivalently a submodule N ≤M is a type submodule if and only ifN isacomplementsubmoduleofM suchthat,forsomesubmoduleC ≤M, N and C do not have nonzero isomorphic submodules and N+C =N⊕C is essential in M. An attempt is made to make this book self contained and accessible to someonewhoeitherhassomeknowledgeofbasicringtheory,suchasbeginning graduate and advanced undergraduate students, or to someone who is willing to acquire the basic definitions along the way. A brief description of the contents of the book is given next. For some readers such a description will only be useful after they have started reading thebook. ThebeginningChapter1definesthemoreorlessstandardnotation used, and lists a few useful facts mostly without proof. Chapter 2 presents allthemoduleclassesthatwillbeused, amongwhicharetorsion,torsionfree classes,σ[M],naturalclasses,andpre-naturalclasses. Chapter3utilizeschain conditions relative to some module class K of right R-modules. These chain conditions guarantee that direct sums of injective modules in K are injective, or that nil subrings are nilpotent in certain rings. Chapter 4 develops the basic theory of type submodules, and the type dimension of a module, which is analogous to the finite uniform dimension. Here new chain conditions, the type ascending and descending chain conditions, are explained and used to obtain structure theorems for modules and rings. Chapter 5 shows that the collection N(R) of all natural classes of right R- modules ordered by class inclusion is a Boolean lattice. By use of this lattice, new natural classes are defined and used to give module decompositions of a module M as a direct sum of submodules belonging to certain natural classes. AmoduleM isaTS-moduleifeverytypesubmoduleofM isadirect summand of M. Extending or CS-modules are a very special case of a TS- module. AdecompositiontheoryforTS-modulesisdevelopedwhichparallels v Copyright 2006 by Taylor & Francis Group, LLC vi the much studied theory of CS-modules. Chapter 6 studies the collection Np(R) of all pre-natural classes of right R-modules. This complete lattice is significant in several ways. First of all, it contains almost all the well and lesser known lattices of module classes as sublattices. Many sublattices of Np(R) are identified. Several connections between ring theoretic properties of the ring R, and purely lattice theoretic properties of Np(R), or some of its many sublattices are proved. Thus, N(R) is a sublattice of Np(R), but in general not a complete sublattice. The interaction between a ring R and the lattices associated to R is explored. If ring theory is to progress, it seems that some new kind of finiteness or chain conditions will be required, such as the type ascending and descending chain conditions. Also, placing restrictive hypotheses on all submodules of a module, or even only on all complement submodules, is too restrictive. However,puttingrestrictionsonthetypesubmodulesonlyismorereasonable. At the time of this writing, this is the only book on the present subject. Previouslyitwasveryinaccessiblyscatteredthroughouttheliterature. More- over, some results in the literature have been either improved or extended, or proofs have been simplified and made more elegant. We view this book not merely as a presentation of a certain theory, but believe that it gives more. It gives tools or new methods and concepts to do ring and module theory. So we regard the book as a program, a path, a direction or road, on which so far we have only started to travel with still a long way ahead. ThesecondauthorexpresseshisgratitudetohiswifeHongwaandtheirson David for their support and patience during this project, and he gratefully acknowledges the support by the Natural Sciences and Engineering Research Council of Canada. John Dauns Yiqiang Zhou Copyright 2006 by Taylor & Francis Group, LLC Note to the Reader Lemmas, corollaries, theorems, etc. are labeled by three integers separated by two periods. Thus (4.2.10) stands for Chapter 4, section 2, and item 10. Chapter 1 and Chapter 2, section 1 (section 2.1) list many facts without proof. In order to read the book, knowledge of the proofs of these facts is not needed. Section 2.1 serves to give an overview of several previously used module classes or categories so that the reader can see where the present material fits into the broader picture. One can read only certain parts of the book without going through every- thing. However,Chapter2,section3onnaturalclassesandChapter4,section 1ontypesubmodulesisbasicforeverythingandshouldnotbeomitted. After reading Chapters 2 and 4, the reader could go anywhere else in the book. vii Copyright 2006 by Taylor & Francis Group, LLC