Classes of modules

222 Pages·2006·1.712 MB·English

by  John Dauns, Yiqiang Zhou

Classes of Modules
Copyright 2006 by Taylor & Francis Group, LLC
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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
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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
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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