Fuzzy Commutative Algebra
This page is intentionally left blank
Fuzzy Commutative Algebra
John N Mordeson D S Malik...
96 downloads
1526 Views
30MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
Fuzzy Commutative Algebra
This page is intentionally left blank
Fuzzy Commutative Algebra
John N Mordeson D S Malik Creighton University
World Scientific Singapore • New Jersey • London • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.
FUZZY COMMUTATIVE ALGEBRA Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.
ISBN 981-02-3628-X
This book is printed on acid-free paper.
Printed in Singapore by U t o P r i n t
To Our Parents
This page is intentionally left blank
vii
Contents FOREWORD
x
PREFACE
xii
LIST OF SYMBOLS
xv
1
1 1 6 10 15 20 29 32 34 36 39
L-SUBSETS A N D L - S U B G R O U P S 1.1 L-Subsets 1.2 L-Subgroups 1.3 Normal L-Subgroups 1.4 Homomorphisms and Isomorphisms 1.5 Complete and Weak Direct Products 1.6 Embedding of Fuzzy Power Sets 1.7 Representation of the Fuzzy Power Algebra 1.8 The Metatheorem 1.9 Applications and Unifications 1.10 EXERCISES
2 L - S U B G R O U P S OF A B E L I A N G R O U P S 2.1 Generators and Direct Sums of L-Subgroups 2.2 Independent Generators 2.3 Primary L-Subgroups 2.4 Divisible and Pure L-Subgroups 2.5 Invariants of L-Subgroups 2.6 Basic and p-Basic L-Subgroups 2.7 EXERCISES
41 41 47 48 50 56 60 68
3 L-SUBRINGS A N D L-IDEALS 3.1 Basic Concepts 3.2 Quotient L-Subrings 3.3 Direct Sums
70 70 83 90
VIII
3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 4
Maximal L-Ideals and Irreducible L-Ideals Prime L-Ideals and Semiprime L-Ideals Radicals of L-Ideals Primary L-Ideals ^-Radicals of L-Ideals 7£-Primary L-Ideals and 7^-Semiprimary L-Ideals Characterization of Artinian Rings L-Generators EXERCISES
L-SUBMODULES 4.1 Basic Concepts 4.2 L-Submodules of Quotient Modules 4.3 L-Submodules Generated by L-Subsets 4.4 Free L-Submodules 4.5 Residual Quotients 4.6 Primary L-Submodules 4.7 7r-Primary L-Submodules 4.8 Primary Decompositions 4.9 EXERCISES
96 100 104 108 Ill 115 119 121 129 131 131 136 139 140 145 149 154 156 160
5 L-SUBFIELDS 5.1 L-Subfields and L-Field Extensions 5.2 Separable and Inseparable Algebraic Extensions 5.3 Composites, Linear Disjointness, and Separability 5.4 Finite-Valued L-Field Extensions 5.5 Separability and Modularity 5.6 Neutrally Closed L-Subfields 5.7 Distinguished L-Subfields 5.8 Splitting 5.9 Purely Inseparable L-Field Extensions 5.10 EXERCISES
161 161 170 175 182 186 189 191 193 198 203
6
205 205 208 216 218 228 230 232 233
S T R U C T U R E OF L - S U B R I N G S A N D L-IDEALS 6.1 Comparison of Radicals 6.2 7£-Primary L-Representations 6.3 ^-Primary Representations 6.4 L-Prime Spectrum of a Ring 6.5 Quasi-local L-Subrings 6.6 Extension of L-Subsets 6.7 Extension of L-Subrings and L-Ideals 6.8 Extension of Prime L-Ideals
ix
6.9 6.10 6.11 6.12 6.13 6.14 6.15
L-Topological Spaces Complete L-Subrings ^Coefficient Fields Existence of L-Coefficient Fields Structure Results Completions EXERCISES
235 240 242 248 252 256 261
7
A L G E B R A I C L-VARIETIES A N D I N T E R S E C T I O N EQUA TIONS 266 7.1 Algebraic L-Varieties 267 7.2 Irreducible Algebraic L-Varieties 275 7.3 Localized L-Subrings 283 7.4 Local Examination 291 7.5 Fuzzy Intersection Equations 296 7.6 L-Intersection Equations 303 7.7 Union Equations 307 7.8 Applications and Examples 309 7.9 EXERCISES 311
8
L-SUBSPACES 8.1 Preliminary Results 8.2 L-Subspaces, //-Subgroups, and L-Subfields 8.3 Nonexistence of Bases 8.4 Existence of Bases 8.5 L-Bases 8.6 Dimension of L-Subspaces 8.7 Existence and Nonexistence of Bases 8.8 Examples 8.9 EXERCISES
315 315 321 329 332 334 337 343 347 353
9
GALOIS THEORY A N D G R O U P L-SUBALGEBRAS 9.1 Galois Theory 9.2 Dimension and Index 9.3 Infinite Fuzzy Galois Theory 9.4 The Galois Correspondence 9.5 Group L-Subalgebras 9.6 Construction of L-Field Extensions 9.7 EXERCISES
355 355 362 364 370 373 376 383
Bibliography
384
Index
401
X
FOREWORD In 1965 I read, with great delight, Lotfi Zadeh's groundbreaking paper "Fuzzy Sets" (Info. Control 8, 338-353). I especially enjoyed Lotfi's "natural" (and elegant!) generalizations of various basic mathematical concepts, starting with the algebra of sets (with n. /Xl§/i2§>.• .•. ®• ^®^n-
t€/
Clearly, if \ii, ^ G L X i with ^ C ^ for each i G / , then
Eh Definition 1.1.13 (Extension Principle) Let f be a mapping from X into Y", and let JJ, e Lx and v G LY. The L-subsets f(/i) G LY and / _ 1 ( ^ ) G L x , defined by Vy G Y, V{^(x) | xex, x € X, f(x) = y} y} t/ /-Hj/) 7^ 0, f(x) =-/ r „ w t A _ / V{/x(x) J/(/*)(