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Fuzzy Commutative
Algebra
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Fuzzy Commutative
Algebra
John N Mordeson D S Malik
Creighton University
World Scientific Singapore • New Jersey • London • Hong Kong
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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 UtoPrint
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To Our Parents
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vii
Contents
FOREWORD x
PREFACE xii
LIST OF SYMBOLS xv
1 L-SUBSETS A N D L-SUBGROUPS 1 1.1 L-Subsets 1 1.2 L-Subgroups 6 1.3 Normal L-Subgroups 10 1.4 Homomorphisms and Isomorphisms 15 1.5 Complete and Weak Direct Products 20 1.6 Embedding of Fuzzy Power Sets 29 1.7 Representation of the Fuzzy Power Algebra 32 1.8 The Metatheorem 34 1.9 Applications and Unifications 36 1.10 EXERCISES 39
2 L-SUBGROUPS OF ABELIAN GROUPS 41 2.1 Generators and Direct Sums of L-Subgroups 41 2.2 Independent Generators 47 2.3 Primary L-Subgroups 48 2.4 Divisible and Pure L-Subgroups 50 2.5 Invariants of L-Subgroups 56 2.6 Basic and p-Basic L-Subgroups 60 2.7 EXERCISES 68
3 L-SUBRINGS A N D L-IDEALS 70 3.1 Basic Concepts 70 3.2 Quotient L-Subrings 83 3.3 Direct Sums 90
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VIII
3.4 Maximal L-Ideals and Irreducible L-Ideals 96 3.5 Prime L-Ideals and Semiprime L-Ideals 100 3.6 Radicals of L-Ideals 104 3.7 Primary L-Ideals 108 3.8 ^-Radicals of L-Ideals I l l 3.9 7£-Primary L-Ideals and 7^-Semiprimary L-Ideals 115 3.10 Characterization of Artinian Rings 119 3.11 L-Generators 121 3.12 EXERCISES 129
4 L-SUBMODULES 131 4.1 Basic Concepts 131 4.2 L-Submodules of Quotient Modules 136 4.3 L-Submodules Generated by L-Subsets 139 4.4 Free L-Submodules 140 4.5 Residual Quotients 145 4.6 Primary L-Submodules 149 4.7 7r-Primary L-Submodules 154 4.8 Primary Decompositions 156 4.9 EXERCISES 160
5 L-SUBFIELDS 161 5.1 L-Subfields and L-Field Extensions 161 5.2 Separable and Inseparable Algebraic Extensions 170 5.3 Composites, Linear Disjointness, and Separability 175 5.4 Finite-Valued L-Field Extensions 182 5.5 Separability and Modularity 186 5.6 Neutrally Closed L-Subfields 189 5.7 Distinguished L-Subfields 191 5.8 Splitting 193 5.9 Purely Inseparable L-Field Extensions 198 5.10 EXERCISES 203
6 STRUCTURE OF L-SUBRINGS A N D L-IDEALS 205 6.1 Comparison of Radicals 205 6.2 7£-Primary L-Representations 208 6.3 ^-Primary Representations 216 6.4 L-Prime Spectrum of a Ring 218 6.5 Quasi-local L-Subrings 228 6.6 Extension of L-Subsets 230 6.7 Extension of L-Subrings and L-Ideals 232 6.8 Extension of Prime L-Ideals 233
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ix
6.9 L-Topological Spaces 235 6.10 Complete L-Subrings 240 6.11 ^Coefficient Fields 242 6.12 Existence of L-Coefficient Fields 248 6.13 Structure Results 252 6.14 Completions 256 6.15 EXERCISES 261
7 ALGEBRAIC L-VARIETIES A N D INTERSECTION EQUATIONS 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 315 8.1 Preliminary Results 315 8.2 L-Subspaces, //-Subgroups, and L-Subfields 321 8.3 Nonexistence of Bases 329 8.4 Existence of Bases 332 8.5 L-Bases 334 8.6 Dimension of L-Subspaces 337 8.7 Existence and Nonexistence of Bases 343 8.8 Examples 347 8.9 EXERCISES 353
9 GALOIS THEORY A N D GROUP L-SUBALGEBRAS 355 9.1 Galois Theory 355 9.2 Dimension and Index 362 9.3 Infinite Fuzzy Galois Theory 364 9.4 The Galois Correspondence 370 9.5 Group L-Subalgebras 373 9.6 Construction of L-Field Extensions 376 9.7 EXERCISES 383
Bibliography 384
Index 401
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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 <, V, A generalizing C, U, fl; but although (or because?) I was then busily engaged in writing a textbook on algebra (An Introduction to Algebraic Structures, Holden-Day, San Francisco, 1968), it didn't occur to me to try my hand at fuzzifying other kinds of mathematical structures.
The idea of trying to fuzzify algebra finally dawned on me several years later, after Lotfi's student C. L. Chang had published his paper "Fuzzy topo-logical spaces" (J. Math. Anal. Appl 24 (1968) 182-190). After somewhat belatedly coming across that paper, I said to myself "If Chang can do it for topological spaces, I can do it for algebraic structures". I took a copy of my own algebra book with me on a train ride from Washington to New York, with the intention of formulating natural fuzzifications of the basic concepts of algebra. Needless to say, I found a way to do this; in fact, by the end of the train ride I had written an essentially complete draft of my embarrassingly well-cited paper "Fuzzy groups" (J. Math. Anal. Appl 35 (1971) 512-517).
As a former algebraist, I take special pleasure in having initiated the study of fuzzy algebraic structures. I regret that I have been unable to contribute to the subject myself since the appearance of my original paper; by the 1970's I had almost entirely abandoned algebra in favor of on assortment of other topics, most of them related to images and their geometric properties. My continued occasional involvement with fuzzy mathematics has expressed itself in non-algebraic directions, deahng primarily with fuzzy graph theory and geometry.
Although I have been unable to participate personally in the vigorous growth of fuzzy algebra, I have derived great enjoyment from observing and encouraging its growth. It therefore gives me special pleasure to welcome the appearance of the first full-length book on the subject. I hope that this happy event is indicative of a new wave of activity in the field, leading both to further
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XI
theoretical developments and to an increased interest in applications. Azriel Rosenfeld
College Park, MD August 1, 1998
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Xli
PREFACE
The notion of a fuzzy subset of a set is due to Lotfi Zadeh. His seminal paper in 1965 has opened up new insights and applications in a wide range of scientific fields. Azriel Rosenfeld used the notion of a fuzzy subset to set down cornerstone papers in several areas of mathematics, among other disciplines. Rosenfeld is the father of fuzzy abstract algebra.
This book is the first to be devoted entirely to fuzzy abstract algebra. The purpose of this book is to give an up to date version of fuzzy commutative algebra. We present a fuzzy ideal theory of commutative rings and apply our results to the solution of fuzzy intersection equations. This treatment includes all the work that has been done on algebraic L-varieties. The book includes all the important work that has been done on L-subspaces of a vector space and on L-subfields of a field. We focus our attention on the connection between L-subgroups of a group and L-subfields of a field. This connection is made in two ways. We present a Galois theory, both finite and infinite, which in the finite case gives a one-to-one inclusion reversing correspondence between finite-valued L-subgroups of the group of automorphisms of a normal field extension and its L-subfields. In the infinite case, we give a Galois correspondence between L-subgroups whose level sets are closed in the Krull topology of the group of automorphisms of a splitting field of a set of separable polynomials over a field. Our second focal point is the connection between L-subgroups of an Abelian group and L-subfields of a field constructed by means of a group algebra. We do not give a complete treatise on L-subgroups. Rather we present results on L-subgroups which are of interest to the connections discussed above.
In Chapter 1, we present the important work of Tom Head and Arthur Weinbereger concerning methods for deriving fuzzy theorems from crisp versions and embedding lattices of fuzzy algebras into lattices of crisp subalgebras. Other than that, we present only material that is needed for the development of the remainder of the book and that may be useful for the further development of fuzzy Galois theory and fuzzy group subalgebras. Since the work of Weinberger is somewhat outside the confines of fuzzy commutative algebra, we only present a summary of it.
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xiii
Essentially only those results concerning L-subgroups of abelian groups which are needed in the development of group L-subalgebras are presented in Chapter 2.
In Chapter 3, we give some basic concepts and properties of L-ideals of a commutative ring with identity. These concepts include those of prime L-ideals, primary L-ideals, and various radicals of L-ideals. Many authors contributed to the development of L-ideal theory. Consequently, we will not attempt to mention them here. Our bibliography lists those who were responsible for the development.
The concepts of quotient L-submodules, free L-submodules, residual quotients, and primary L-ideals are given Chapter 4. We do not give a complete treatment of L-submodules, but we do give an extensive bibliography on L-submodules. However we stop short of including papers which fall into the realm of category theory.
Chapter 5 deals with L-subfields and L-field extensions. Concepts such as separability and inseparability for L-field extensions are developed. We introduce a concept for L-field extensions which is not present in ordinary field extensions, namely that of neutral L-field extensions.
In Chapter 6, we delve deeper infco the realm of L-subrings. We present a primary representation theory for L-ideals which is used in the next chapter to examine fuzzy intersection equations. Other topics in this chapter include the L-prime spectrum of a ring, quasi-local L-subrings, and L-coefficient fields. The results of this chapter are mostly those of the authors.
In Chapter 7, we give a complete account of algebraic L-varieties and fuzzy intersection equations. This material is entirely that of the authors. We also examine fuzzy intersection equations locally.
In Chapter 8, we give an axiomatic approach to the concept of the generation of L-algebraic substructures. We determine the generating properties which yield the underpinnings for the existence of free generating sets and the uniqueness of their cardinalities for L-algebraic substructures. These L-substructures include concepts of L-subspaces of vector spaces, relative p-bases for L-subfields, transcendence bases for L-subfields, and L-subgroups in terms of p-bases and torsion-free bases. Many authors contributed to the development of L-subspaces. They are included in our bibliography. We do not include in the bibliography papers dealing with topological vector spaces.
In Chapter 9, we make a connection between L-subgroups and L-subfields in two ways. First, we make the connection by means of Galois theory and then by means of group algebras. Here also the work is entirely that of the authors.
We assume that the reader is familiar with the basic results and definitions from abstract algebra and lattice theory.
The authors are grateful to the staffs of World Scientific Publishers, es-
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xiv
pecially Sen Hu and S. H. Gan. We are indebted to Pr. Michael Proterra, Dean, Creighton College of Arts and Sciences and to Dr. and Mrs. George Haddix for their support of our work. We benefited from the work of Professor Yu Yandong during his visit. We also wish to thank Professor Paul Wang of Duke University for his support of fuzzy mathematics. The first author dedicates the book to his grandchildren, John, Mary Pat, Michael, Christine Marie, Matthew, Marc, and Jack. The second author dedicates the book to his daughter Shelly Malik, who is a joy and her smile is what counts.
John N. Mordeson D. S. Malik
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XV
LIST OF SYMBOLS
A V N Z Q R 0 I L e t c c D D A\B Lx
[o,i]x
M(X) Im(/i) M* Ma P(S) u n foot(<S) a\b a / 6 o
n ft i € / /(*») / _ 1 ( " )
minimum or infimum maximum or supremum the set of positive integers the set integers the set of rational numbers the set of real numbers the empty set an arbitrary nonempty index set complete Heyting algebra belongs to does not belong to subset proper subset contains properly contains set difference L-power set of X fuzzy power set of X fJ,(X) = {//(x) | x € X } , the image of /x Im(/x) = {fJ>(x) | x G X } , the image of /i fj,* = {x \ x £ X,ii(x) > 0}, the support of \x fj,a = {x | x € X,fj,(x) > a} , a-cut or a-level set of /x power set of the set S union of sets intersection of sets foot(<S) = {y | ya € 5 } a divides b a does not divide 6 composition product
complete direct product
image of \i under / the pre-image or inverse image of v under /
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xvi
/ i O I /
M-1
L(G) M. <M> NL(G) NO*) \1<V G/ii
v/fi r^>
~ « =
n iei
n C(X) 9* £ (F) L(p) ^(/*) Ji + V \X — V
-M L(fl) L/(.R) ju M1
/*n
M « /«W <M> R/n E* ^c
V? W F(J>) fj. : v
M:C L(F)
the product of \x and 1/ the inverse of fi the set of all L-subgroups of G /x* = { i G G | JJL(X) = /x(e)}> where G is a group the L-subgroup of G generated by /i the set of all normal L-subgroups of G the normalizer of /i in G [i is a normal L-subgroup of the L-subgroup v the quotient group or factor group of G relative to the normal L-subgroup fi the quotient L-subgroup weakly homomorphic weakly isomorphic homomorphic isomorphic
weak product
weak direct product
the crisp power set of X representation function the characteristic function of Y L(JJ) = {vG L{G) \vQfi «F(M) = {veLG \vCfi} the sum of /x and v the difference of \i and v the negative of fi the set of all L-subrings of R the set of all L-ideals of R /z* = {x £ R | fj.(x) = ^(0)}) where R is a ring / i 1 = / i /xn = / i 1 o / i n - 1
^ > = / i //"> = / /v—1) •the L-ideal of i? generated by // the quotient ring of R by /i weak direct sum the family of all prime L-ideals fj, of R such that £ C /z and £* C //* L-radical of £ 1l(/i)(x) = VneNM(zn)V i e i ? , the ^-radical of fi HP) = {C € L* | c c P} fi : v = U{77 | 77 G LH , 77 • 1/ C / i}, residual quotient /i : C = U{f | ^ L M , ( ^ C / i } , residual quotient the set of all L-subfields of F
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v/n L(u) ^ M# QM Kdl(fi) X v(x) L-Spec(R) Fspec(-R) (R,M) R# M{n) FS{V)
n») v? VS(n)
°# L(F) L(F/K) AK
C{F/K)
K[G) [a:K] [F:<x] H [A: n] L(\) L(\/K)
s(0 ■
xvii
L-field extension L{u) = {M € L(F) 1/xC./} neutral closure of an L-subfield \i fi# = {x€R\ fi(x) > / i(l)} Qn = {T)\T) is a £-prime L-ideals of i2 such that 77 D /x} fcd/(/x) = D r ^ Q ^ 3E = {77 | 77 is a prime L-ideal of R} V(x) = iv e x | x c r,} L-prime spectrum the fuzzy prime spectrum complete local ring R with unique maximal ideal M set of nonnegative real numbers algebraic L-variety of /x the set of all sets S of L-singletons of V such that if xa,Xb € <S, then a = b > 0 ^(/x) = {X G L v | X C /i} V ^ / T ^ a , ! ] ) P5(/x) = {<S | <S e FS(V) , Vza € 5 , x a C /x}. ( 7 # = { l G F | d(x) > (7(1)} the set of all L-subfields of F L(F/K) = {<T\ a € L(F) and a # D K } the set of all automorphisms of F which are the identity on K the set of all finite-valued L-subfields a of F such that Gb 2 K where b is the maximal element in Im(cr) group algebra fuzzy dimension of a over K fuzzy dimension of F over a fuzzy order of /x fuzzy index of /x in A L(X) = {K € L(F)\ K C A } L(X/K) = {4> € J"(F) | / c C « / ) C A } 5 ( 0 = { * « ! * € £ • , « * ) = <*} end of proof
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