Lattice Concepts of Module Theory

232 Pages·2000·8.624 MB·English

by  Grigore Călugăreanu (auth.)

Lattice Concepts of Module Theory
Kluwer Texts in the Mathematical Sciences
VOLUME 22
A Graduate-Level Book Series
Lattice Concepts of
Module Theory
by
Grigore Calugareanu
Department ofA lgebra,
Faculty of Mathematics and Computer Sciences,
Babq-Bolyai University,
Cluj-Napoca, Romania
Springer-Science+Business Media, B.V.
A c.I.P. Catalogue record for this book is available from the Library of Congress.
Printed on acid-free paper
All Rights Reserved
ISBN 978-90-481-5530-9 ISBN 978-94-015-9588-9 (eBook)
DOI 10.1007/978-94-015-9588-9
© 2000 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 2000.
Softcover reprint of the hardcover I st edition 2000
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
This volume is dedicated to the memory of my father George.
Contents
Preface ix
List of Symbols xi
1 Basic notions and results 1
2 Compactly generated lattices 17
3 Composition series. Decompositions 29
4 Essential elements. Pseudo-complements 39
5 Socle. Torsion lattices 47
6 Independence. Semiatomic lattices 59
7 Radical. Superfluous and fully invariant elements 77
8 Lattices of finite uniform dimension 87
9 Purity and neatness in lattices 99
10 Coatomic lattices 115
11 Co-compact lattices 123
12 Supplemented lattices. Locally artinian lattices 131
13 Several dimensions 143
CONTENTS
Vlll
14 Solutions of exercises 165
Bibliography 219
Index 223
Preface
It became more and more usual, from, say, the 1970s, for each book
on Module Theory, to point out and prove some (but in no more than
15 to 20 pages) generalizations to (mostly modular) lattices. This was
justified by the nowadays widely accepted perception that the structure
of a module over a ring is best understood in terms of the lattice struc
ture of its submodule lattice. Citing Louis H. Rowen "this important
example (the lattice of all the submodules of a module) is the raison
d'etre for the study of lattice theory by ring theorists". Indeed, many
module-theoretic results can be proved by using lattice theory alone.
The purpose of this book is to collect and present all and only the
results of this kind, although for this purpose one must develop some
significant lattice theory.
The results in this book are of the following categories: the folklore
of Lattice Theory (to be found in each Lattice Theory book), module
theoretic results generalized in (modular, and possibly compactly gen
erated) lattices (to be found in some 6 to 7 books published in the last
20 years), very special module-theoretic results generalized in lattices
(e.g., purity in Chapter 9 and several dimensions in Chapter 13, to be
found mostly in [27], respectively, [34] and [18]) and some new con
cepts (e.g., fully invariant elements, CS (extending) and FI-extending
lattices) introduced in lattices by the author (mostly adding the so
called H-noetherian and restricted so de conditions to the previous al
ready mentioned conditions).
Almost all the results in the first two categories are well-known in
lattice theory. They appear here with their (detailed) proofs in order
to make this work self-contained. Some might say that the proofs are
too detailed. Indeed, it was the author's deliberate intention to write
IX
x PREFACE
a book which would be easy to read.
Most of the examples are chosen (as in [41] or [17]) from abelian
groups, considering that these are the most simple modules and that
everybody knows the fundamental results of Abelian Group Theory.
Abelian Group Theory is clearly the violon d'Ingres of the author.
It seems that lately it has not received very much attention, everybody
working in Module Theory. Now situations of the following kind occur:
for new module concepts (not arisen from Abelian Group Theory) the
abelian group examples for these concepts are not fully known. Some
of them are treated as exercises.
Chapter 13, concerning the several dimensions one can define in
lattices is very much like the corresponding one from [34]. The author
has included it in order to make this book as self-contained as possible.
The Chapter with the solutions of the exercises was included with
the same idea, to simplify the reader's task.
This is not a treatise on Lattice Theory. It clearly does not contain
the genuine problems of Lattice Theory. This is actually a book which
helps one to study Module Theory after learning all the more general
results one can prove in lattices with reasonable conditions.
While this book is written as a textbook, I hope it will be also useful
at a professional level.
Finally, permit me to quote "all the good results belong to others,
all the mistakes are mine".
List of Symbols
Symbol Description
p the set of the prime numbers
N the set of the non-negative integer numbers
N* = N - {O} the set of the positive integer numbers
z
the set of the integer numbers
Q the set of the rationals numbers
R the set of the real numbers
C the set of the complex numbers
D the dyadic numbers
P(M) the set of the subsets of M
Po(M) the set of the finite subsets of M
im(f) the image of f
ker(J) the kernel of f
f-l(y) the pre-image of Y by f
R-1 the inverse relation of R
R(A) a section of a relation R
SR(M) all the submodules of RM
(A,O) an universal algebra
S(A,O) all the subalgebras of (A, 0)
'R all the partial order relations on a given set
< a partial ordering
VX the join of the elements of X
AX the meet of the elements of X
aVb=sup(a,b) the smallest upper bound
a /\ b = inf(a, b) the largest lower bound
a-<b the element a is covered by b
allb non-comparable elements
Xl