Fuzzy commutative algebra and intersection equations

Information Sciences 128 (2000) 127±145

www.elsevier.com/locate/ins

Fuzzy commutative algebra and intersection equations John N. Mordeson

*

Department of Mathematics and Computer Science, Creighton University, 2500 California Plaza, Omaha, NE-68178, USA Received 1 June 1999; accepted 28 April 2000

Abstract In this paper, we discuss the status of fuzzy commutative algebra as it applies to intersection equations. We give an up to date account of the theory of nonlinear systems of intersection equations of L-singletons. The concept of an algebraic L-variety is presented in order to bring fuzzy commutative algebra to bear on the solution of such intersection equations. Ó 2000 Elsevier Science Inc. All rights reserved. Keywords: Algebraic L-varieties; Intersection equations; L-ideal; R-primary L-representations

1. Introduction Unless otherwise speci®ed, L is a complete chain throughout this paper. We let 1 and 0 denote the maximal and minimal elements of L; respectively. We assume that the reader is familiar with the de®nitions and basic results of fuzzy commutative ring theory. Since the seminal paper of Rosenfeld [17], the theory of fuzzy abstract algebra has developed rapidly. However, it remains in its infancy. Some areas of fuzzy abstract algebra lack direction. The area with the most direction and purpose is fuzzy commutative algebra. Its results can be applied to nonlinear

*

Fax: +402-280-5758. E-mail address: [email protected] (J.N. Mordeson).

0020-0255/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 0 - 0 2 5 5 ( 0 0 ) 0 0 0 4 1 - 4

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systems of intersection equations of L-singletons. The purpose of this paper is to give an up to date account of the theory of nonlinear systems of intersection equations of L-singletons. We bring fuzzy commutative algebra to bear on a natural application area, namely, the solution of such intersection equations. Let R denote the polynomial ring F ‰x1 ; . . . ; xn Š, where F is a ®eld and x1 ; . . . ; xn are algebraically independent indeterminants over F. Let C be a ®eld containing F. C may be taken to be the algebraic closure of F or an algebraically closed ®eld with in®nite transcendence degree over F. Let C k denote the set of all ordered k-tuples with entries from C; k a positive integer. Our approach is to consider those L-ideals l of R which are ®nite-valued and are such that l…0† ˆ 1 since these are precisely the L-ideals of R which have R-primary L-representations [6,8] and which represent nonlinear systems of equations of L-singletons [13]. We de®ne the algebraic L-variety M…l† of l and show that from an irredundant R-primary L-representation of l; M…l† is a ®nite union of irreducible algebraic L-varieties, no one of which is contained in the union of the others. We then apply this result to the solution of a nonlinear system of intersection equations of L-singletons. We show that there exists an L-ideal l of R which represents this system and the irredundant R-primary L-reprep sentation of l displays the solution of the system in a manner similar to that of the crisp situation. 2. Algebraic L-varieties If I is an ideal of R; we let M…I† denote the algebraic variety of I; [9, p. 203]. If Z is a subset of C k ; we let I…Z† denote the set of all f 2 R which vanish at all points of Z. Then I…Z† is an ideal of R; [9, p. 203]. We now give de®nitions for the fuzzy counterparts of M and I. Let c be a strictly decreasing function of L into itself such that c…0† ˆ 1; c…1† ˆ 0; and 8a 2 L; c…c…a†† ˆ a. The following approach has the advantage that c may be changed to ®t the application. The proofs of this section can be found in [10,11]. De®nition 1. Let v be a fa0 ; a1 ; . . . ; an g where a0 < a1 follows: 8 < c…an † if f I…v†…f † ˆ c…ai † if f : c…a0 † if f

®nite-valued L-subset of C k ; say Im…v† ˆ <


< an . De®ne the L-subset I…v† of R as 2 R n I…van †; 2 I…vai‡1 † n I…vai †; 2 I…va1 †:

i ˆ 1; . . . ; n ÿ 1;

If n ˆ 0, then we de®ne I…v†…0† ˆ 1. De®nition 2. Let l be a ®nite-valued L-ideal of R; say Im…l† ˆ fb0 ; b1 ; . . . ; bm g where b0 < b1 <


< bm . De®ne the L-subset M…l† of C k as follows:

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8 > < c…bm † M…l†…z† ˆ c…bi † > : c…b0 †

129

if z 2 C k n M…lbm †; if z 2 M…lbi‡1 † n M…lbi †;

i ˆ 1; . . . ; m ÿ 1;

if z 2 M…lb1 †:

M…l† is called an algebraic L-variety (of l). In De®nition 1, it is possible for I…vai‡1 † ˆ I…vai † or R ˆ I…van †. In this case c…ai † 62 Im…I…v††; i ˆ 1; . . . ; n. Similarly, it is possible for c…bi † 62 Im…M…l†† for some i ˆ 1; . . . ; m. Proposition 3. Let v be defined as in Definition 1. Then 1. I…v†c…ai † ˆ I…vai‡1 † for i ˆ 0; 1; . . . ; n ÿ 1; 2. if 0 6 b 6 c…an †; then I…v†b ˆ R; 3. if c…ai‡1 † < b < c…ai †; then I…v†b ˆ I…vc…b† † for i ˆ 0; 1; . . . ; n ÿ 1; 4. if c…a0 † < b 6 1; then I…v†b ˆ ;. For v as de®ned in De®nition 1, I…v†c…ai † ˆ I…vai‡1 † is an ideal of R for i ˆ 0; 1; . . . ; n ÿ 1. Thus, since Im…v†  fc…ai † j i ˆ 0; 1; . . . ; ng; I…v† is a L-ideal of R. Proposition 4. Let l be defined as in Definition 2. Then 1. M…l†c…bi † ˆ M…lbi‡1 † for i ˆ 0; 1; . . . ; m ÿ 1; 2. if 0 6 a 6 c…bm †; then M…l†a ˆ C k ; 3. if c…bi‡1 † < a < c…bi †; then M…l†a ˆ M…lc…a† † for i ˆ 0; 1; . . . ; m ÿ 1; 4. if c…b0 † < a 6 1; then M…l† ˆ ;. Proposition 5. Let v and l be as defined on Definitions 1 and 2, respectively. Then 1. jIm…M…I…v††j ˆ jIm…I…v††j; 2. jIm…I…M…l††j ˆ jIm…M…l††j. Proposition 6. Let v and l be defined as in Definitions 1 and 2, respectively. Then 1. 8a 2 L; M…I…v††a ˆ M…I…va ††; 2. 8b 2 L; I…M…l††b ˆ I…M…lb ††. Proposition 7. Let v and l be defined as in Definitions 1 and 2, respectively. Then 1. I…M…I…v††† ˆ I…v†; 2. M…I…M…l††† ˆ M…l†. Theorem 8. Let a be an L-subset of C k . Then a is an algebraic L-variety if and only if a is finite-valued and 8a 2 Im…a†; aa is an algebraic variety.

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Proposition 9. If l is a nonconstant prime L-ideal of R; then l ˆ I…M…l††:

p Let l be an L-ideal of R. Then the radicals l and R…l† are defined in [14, Definition 3.6.1, p. 104, Definition 3.8.1, p. 111], respectively. Theorem 10. Suppose that l is defined as in Definition 2 and l…0† ˆ 1. Then p M…l† ˆ M… l† ˆ M…R…l††. Corollary 11. Let L be a dense chain. If p is a prime L-ideal of R belonging to the primary L-ideal w of R; then M…w† ˆ M…p†. Lemma 12. Suppose that l and m are L-ideals of R such that Im…l† ˆ fb0 ; b1 ; . . . ; bm g where b0 < b1 <


< bm ˆ 1 and Im…m† ˆ fb; 1g where b < 1. Then M…l \ m† ˆ M…l† [ M…m†. Theorem 13. If l and m are finite-valued L-ideals of R such that l…0† ˆ m…0† ˆ 1; then M…l \ m† ˆ M…l† [ M…m†. Lemma 14. Let v and g be finite-valued L-subsets of a set S. If vb ˆ gb 8b 2 ‰0; 1†; then v1 ˆ g1 . Lemma 15. Suppose that a and b are algebraic L-varieties such that Im…a† ˆ fa0 ; a1 ; . . . ; an g where 0 ˆ a0 < a1 <


< an and Im…b† ˆ f0; ag where 0 < a 6 an . Then I…a [ b† ˆ I…a† \ I…b†. Theorem 16. If a and b are algebraic L-varieties such that 0 2 Im…a† \ Im…b†; then I…a [ b† ˆ I…a† \ I…b†. 3. Irreducible algebraic L-varieties De®nition 17. Let a be an algebraic L-variety. Then a is irreducible if 8 algebraic L-varieties n and f such that a ˆ n [ f either a ˆ n or a ˆ f; otherwise a is called reducible. Theorem 18. Let a be an algebraic L-variety. Then a is irreducible and nonconstant if and only if Im…a† ˆ f0; ag; 0 < a; and aa is irreducible. Theorem 19. Let l be a nonconstant finite-valued L-ideal of R. Then I…M…l†† is prime if and only if M…l† is irreducible. Theorem 20. Let l be a finite-valued L-ideal of R with l…0† ˆ 1. Then p I…M…l†† ˆ l ˆ R…l†.

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Since this section deals with L-ideals l which are ®nite-valued and have the property that l…0† ˆ 1; it is evident by now that the results in this section hold p interchangeably with l and R…l†. Corollary 21. Let l and m be finite-valued L-ideals of R such that l…0† ˆ p p m…0† ˆ 1. Then M…l†  M…m† if and only if l  m. Theorem 22. There exists a one-to-one correspondence between algebraic L-varieties a with 0 2 Im…a† and radical L-ideals. Theorem 23. Every algebraic L-variety a with 0 2 Im…a† can be uniquely expressed as the union of a finite number of irreducible algebraic L-varieties no one of which is contained in the union of the others. p Given a ®nite-valued L-ideal l of R with l…0† ˆ 1. Let l ˆ p1 \


\ pr be a irredundant R-primary L-representation. Then the pi are the minimal prime L-ideals belonging to l. We may thus obtain M…l† as the union of algebraic L-varieties of the minimal prime L-ideals among the prime L-ideals belonging to the R-primary L-ideals in an irredundant R-primary L-representation of l. Example 24. Let L ˆ ‰0; 1Š. Let R ˆ F ‰x; y; zŠ where F is the ®eld of complex numbers and x; y; z are algebraically independent indeterminants over F. De®ne the fuzzy subset l of R by 8 1 if f ˆ 0; > > > < 1 if f 2 hx2 zi n h0i; l…f † ˆ 21 2 2 2 2 > > > 4 if hx ‡ y ÿ 1; x zi n hx zi; : 2 2 0 if f 2 R n hx ‡ y ÿ 1; x2 zi: p Then l is a fuzzy ideal of R. Now l is such that 8 1 if f ˆ 0; > > > < 1 if f 2 hxzi n h0i; p l…f † ˆ 21 > if hx2 ‡ y 2 ÿ 1; xzi n hxzi; > > :4 0 if f 2 R n hx2 ‡ y 2 ÿ 1; xzi: Hence l0 ˆ R; l1=4 ˆ hx2 ‡ y 2 ÿ 1; x2 zi; l1=2 ˆ hx2 zi; l1 ˆ h0i;

p l0 ˆ R; p l1=4 ˆ hx2 ‡ y 2 ÿ 1; xzi; p l1=2 ˆ hxzi; p l1 ˆ h0i:

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Since F k ˆ M…h0i†; 8 1 c… † > > < 2 M…l†…w† ˆ c…14† > > : c…0† ˆ 1

if w 2 M…h0i† n M…hxzi†; if w 2 M…hxzi† n M…hx2 ‡ y 2 ÿ 1; xzi†; if w 2 M…hx2 ‡ y 2 ÿ 1; xzi†:

Consider the fuzzy subsets x; v; g of R de®ned by  1 if f 2 hx2 ‡ y 2 ÿ 1; x2 zi; x…f † ˆ 0 otherwise:  1 if f 2 hx2 zi; v…f † ˆ 1 otherwise: 4  1 if f 2 h0i; g…f † ˆ 1 otherwise: 2 Then x; v; g are fuzzy ideals of R and l ˆ x \ v \ g. De®ne the fuzzy subsets w…i† of R; i ˆ 1; . . . ; 6 by  1 if f 2 hx2 ; y ÿ 1i; …1† w …f † ˆ 0 otherwise: …2†



w …f † ˆ …3†



w …f † ˆ …4†



w …f † ˆ …5†

w …f † ˆ

if f 2 hx2 ; y ‡ 1i; otherwise:

1 0

if f 2 hx2 ‡ y 2 ÿ 1; zi; otherwise:

1

if f 2 hx2 i; otherwise:

1

if f 2 hzi; otherwise:

1

if f 2 h0i; otherwise:

1 4



w …f † ˆ …6†

1 0

1 4



1 2

Then w…i† is a fuzzy ideal of R; i ˆ 1; . . . ; 6 such that x ˆ w…1† \ w…2† \ w…3† , since hx2 ‡ y 2 ÿ 1; x2 zi ˆ hx2 ; y ÿ 1i \ hx2 ; y ‡ 1i \ hx2 ‡ y 2 ÿ 1; zi; v ˆ w…4† \ w…5† since hx2 zi ˆ hx2 i \ hzi;

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g ˆ w…6† . Thus l ˆ \6iˆ1 w…i† and in fact this is an irredundant fuzzy primary represntation of l. Now  q 1 if f 2 hx; y ÿ 1i; …1† w …f † ˆ 0 otherwise:  q 1 …2† w …f † ˆ 0

if f 2 hx2 ; y ‡ 1i; otherwise:

q w…3† ˆ w…3† ;  q 1 w…4† …f † ˆ 1

if f 2 hxi; otherwise:

4

q w…5† ˆ w…5† ; q w…6† ˆ w…6† : q p 6 …i† …i† Hence l ˆ \iˆ1 p , where p ˆ w…i† is a fuzzy prime ideal of R; i ˆ 1; . . . ; 6. We have the following fuzzy algebraic varieties:  M…p…1† †…w† ˆ  M…p…2† †…w† ˆ  M…p…3† †…w† ˆ  M…p…4† †…w† ˆ  M…p…5† †…w† ˆ

1

if w 2 M…hx; y ÿ 1i†;

0

otherwise:

1 0

if w 2 M…hx; y ‡ 1i†; otherwise:

1

if w 2 M…hx2 ‡ y 2 ÿ 1; zi†;

0

otherwise:

c…14† 0

if w 2 M…hxi†; otherwise:

c…14†

if w 2 M…hzi†;

0

M…p…6† †…w† ˆ c…12†

otherwise: 8w 2 C k :

Then M…l† ˆ [6iˆ1 M…p…i† † and in fact M…p…i† † is irreducible and no M…p…i† † is contained in the union of the others, i ˆ 1; . . . ; 6. Consider the nonlinear system of equations of fuzzy singletons, i.e. L-singletons, …xb †2 ‡ …ya †2 ÿ 11=4 ˆ 01=4 ;

…xb †2 zu ˆ 01=2 :

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Then a solution is given by a P 14 and b ^ u ˆ 12 and the solution of x2 ‡ y 2 ÿ 1 ˆ 0 and x2 z ˆ 0. Note also that l ˆ h…x2 ‡ y 2 ÿ 1†1=4 ; …x2 z†1=2 i. If we let c…0† ˆ 1;

c…14† ˆ 12;

c…12† ˆ 14;

c…1† ˆ 0;

then the above representation of M…l† seems to better represent the solution of the above nonlinear system of equations of fuzzy singletons. The M…p…i† † for i ˆ 1; 2; 3; yield the crisp part of the solution, while M…p…i† † for i ˆ 4; 5; 6 yield the fuzzy part. In Example 24, it was shown how a solution to a system of fuzzy intersection equations could be displayed by a primary representation of the L-ideal generated by the de®ning polynomials of the intersection equations. We now show this holds in general. The proofs of the results are in [13]. Theorem 25. Let l ˆ h…f1 †a1 ; . . . ; …fq †aq i [ 1f0g where f1 ; . . . ; fq 2 R; 1 P a1 P


P aq > 0 and a 6ˆ aq . Suppose that hf1 ; . . . ; fq i 6ˆ R. Let fai1 ; . . . ; aim g ˆ fa1 ; . . . ; aq g be such that ai1 >


> aim . Let Famÿuÿ1 ˆ ffk j ak > aimÿu g; u ˆ 0; 1; . . . ; m ÿ 1; and let Faim ˆ ff1 ; . . . ; fq g. Define x; x1 ; . . . ; xm of R as follows: ( 1 if r 2 hFaim i; x…r† ˆ 0 if r 62 hFaim i; ( 1 if r 2 hFaimÿu i; u ˆ 1; . . . ; m ÿ 1: xu …r† ˆ aimÿu‡1 if r 62 hFaimÿu i; ( 1 if r 2 h;i; xm …r† ˆ ai1 if r 62 h;i:

the

L-subsets

Then x; x1 ; . . . ; xm are L-ideals of R and l ˆ x \ x1 \


\ xm . Theorem 26. Let l ˆ h…f1 †a1 ; . . . ; …fq †aq i [ 1f0g where f1 ; . . . ; fq 2 R; 1 P a1 P


P aq > 0 and a1 6ˆ aq . Suppose that hf1 ; . . . ; fq i 6ˆ R. Let x; x1 ; . . . ; xm be defined as in Theorem 25. Let 1x ˆ Q01 \


\ Q0k0 and 1xu ˆ Qu1 \


\ Quku be R-primary representations of 1x and 1xu ; respectively, u ˆ 1; . . . ; m. For each u ˆ 0; 1; . . . ; m; define the L-subsets lu1 ; . . . ; luku of R as follows: 8r 2 R;

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 l0j …r† ˆ  luj …r† ˆ

1 0

if r 2 Q0j ; if r 2 6 Q0j ;

j ˆ 1; . . . ; k0 ;

1 0

if r 2 Quj ; if r 2 6 Quj ;

j ˆ 1; . . . ; ku ; u ˆ 1; . . . ; m:

135

Then the following assertions holds: 1. lu1 ; . . . ; luku are L-ideals of R; u ˆ 0; 1; . . . ; m. 2. xu ˆ lu1 \


\ luku ; u ˆ 0; 1; . . . ; m. 3. l ˆ …l01 \


\ l0k0 † \ …l11 \


\ l1k1 † \


\ …lm1 \


\ lmkm † is an Rprimary L-representation of l. Let R denote the polynomial ring F ‰x1 ; . . . ; xn Š in n indeterminates over the ®eld F. Then every ideal of R has a primary representation. Let k1j X






i1 ˆ1

knj X in ˆ1

 i1  in …ri1


in j †1 …x1 †b1j


…xn †bnj ˆ …rj †aj ;

j ˆ 1; . . . ; q;

…1†

denote q nonlinear equations in the L-singletons …x1 †b1 ; . . . ; …xn †bn where bij ˆ bi if xi appears in equation j and 1 otherwise, i ˆ 1; . . . ; n; j ˆ 1; . . . ; q and where the …ri1


in j †1 and the …rj †aj are L-singletons and the rj and the ri1


in are in F. Let fj ˆ

k1j X i1 ˆ1






knj X in ˆ1

i

i

ri1


in j …x1 † 1


…xn † n ;

j ˆ 1; . . . ; q:

Then the system of equations (1) is equivalent to the following two systems of equations: f j ˆ rj ;

j ˆ 1; . . . ; q

…2†

and b1j ^


^ bnj ˆ aj ;

j ˆ 1; . . . ; q:

…3†

Let l ˆ h…f1 †a1 ; . . . ; …fq †aq i [ 1f0g . It is clear that in (3) of Theorem 26, p p l01 \


\ l0k0 gives, via unions of the corresponding irreducible algebraic L-varieties, the crisp part (2), of the solution to the L-intersection equations (1), while ÿp ÿp p
p
l11 \


\ l1k1 \


\ lm1 \


\ lmkm gives the fuzzy part (3).

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4. Localized L-subrings The notion of algebraic L-varieties was introduced in order to use primary representation theory of L-ideals to examine the solution of fuzzy intersection equations. The concepts of quasi-local L-subrings and complete local L-subrings were developed in [7] and [4,12], respectively, in order to lay the ground work for the examination of fuzzy intersection equations locally. In this section, we characterize local rings in terms of certain L-ideals. We also characterize rings of fractions at a prime ideal in terms of L-ideals. We apply our results to fuzzy intersection equations. In particular, we show that the L-ideal which represents a system of fuzzy intersection equations in a polynomial ring is such that its extension in a ring of fractions represents the same system of fuzzy intersection equations. The results in this section are taken from [1]. Throughout this section R denotes a commutative ring with identity and L is a dense chain. Let l ˆ fx 2 R j l…x† ˆ l…0†g and l# ˆ fx 2 R j l…x† > l…1†g. Then l and l# are ideals of R. Let S be a set of L-singletons of R such that if xa ; xb 2 S; then a ˆ b > 0. Let foot…S† ˆ fx j xa 2 Sg. If l is an L-ideal of R such that l ˆ hSi [ 0l…0† for some S; then S is called a generating set for l. If S is a generating set for l; and hS n fxa gi [ 0l…0†  l 8xa 2 S; then S is called a minimal generating set for l. If S is a subset R; we let hSi denote the ideal of R generated by S. A commutative ring with identity, but not necessarily Noetherian, is said to be local if it has a unique maximal ideal. (Such a ring is called quasi-local in [7]). In [7] the de®nition of a quasi-local L-subring of R was given when R was assumed to be local. That is, an L-subring l of a local ring R was called quasilocal if l…x† ˆ l…xÿ1 † for all units x of R. If l is an L-ideal of R; then l…x† ˆ l…1† for all units x of R. Hence if R is a local ring and l is an L-ideal of R; then l is a quasi-local L-subring of R. We also know that if l is an L-ideal of R; then l…y† P l…1† 8y 2 R. If l is a nonconstant L-ideal of R; then l…0† > l…1†. De®nition 27. An L-ideal l of R is called local if 8x 2 R; l…x† ˆ l…1† is equivalent to x being a unit in R. Note that if l is an L-ideal of R which is local, then l is not constant since 0 is not a unit of R. Let R denote the polynomial ring F ‰xŠ over the ®eld F. De®ne the L-subring l of R by l…z† ˆ 1 if z ˆ 0; l…z† ˆ 12 if z 2 F n f0g; and l…z† ˆ 14 if z 2 R n F . Then l is an L-subring of R. Also, l…z† ˆ l…1† if and only if z is a unit. However, l is not an L-ideal of R. We also note that R is not a local ring. Lemma 28. Let l be a nonconstant L-ideal of R. Then l is local if and only if l# is the unique maximal ideal of R.

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Recall that an L-ideal l of R is a generalized maximal L-ideal if l is not constant and for any L-ideal m of R; if l  m; then either l ˆ m or m ˆ 1R . Then an L -ideal l of R is maximal if and only if jIm…l†j ˆ 2; l…0† ˆ 1; and l is a maximal ideal of R. Let l and k be L-ideals of R. Then l and k are said to be equivalent if fla j a 2 Im…l†g ˆ fka j a 2 Im…k†g: Theorem 29. The following conditions are equivalent: 1. R is local, 2. R has an L-ideal which is local, 3. all generalized maximal L-ideals of R are local, 4. all generalized maximal L-ideals of R are equivalent. If R is Artinian, we say that an L-ideal l of R is of maximal chain if the level ideals of l form a composition series. Theorem 30. Let R be Artinian. Then R is local if and only if every L-ideal of R of maximal chain is local. An L-ideal l of R is called normalized if l…0† ˆ 1. Theorem 31. R is a field if and only if the set of all normalized L-ideals of R which are local coincides with the set of all generalized maximal L-ideals of R. Throughout the remainder of the section, S denotes a closed multiplicative system in R such that 0 62 S and which is saturated, i.e., 8x; y 2 R; xy 2 S implies x; y 2 S [3]. Let RS ÿ1 denote the corresponding ring of fractions. Then RS ÿ1 ˆ f/…r†=/…s† j r 2 R; s 2 Sg, where / is a homomorphism of R into RS ÿ1 such that Ker / ˆ fx 2 R j xs ˆ 0 for some s in Sg and the elements of /…S† are units in RS ÿ1 [18, p. 222]. If I is an ideal of R; we use the notation IS ÿ1 for the ideal of RS ÿ1 generated by /…I†. De®nition 32. Assume l and l0 are L-ideals of R and RS ÿ1 ; respectively. Then l0 is called the localized L-subring of l in RS ÿ1 if Im…l† ˆ Im…l0 † and l0a ˆ la S ÿ1 8a 2 Im…l†. In the following example, we show that not every L-ideal of R has a localized L-subring in RS ÿ1 . We say that the ring of fractions RS ÿ1 is a localized ring of R at a prime ideal, if there exists a prime ideal P of R such that S ˆ cP ; the complement of P in R.

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Theorem 33. The ring of fractions RS ÿ1 is a localized ring of R at a prime ideal of R if and only if there exists an L-ideal l of R which has a localized L-subring l0 in RS ÿ1 and S  R n l# . In such a case, l# is a prime ideal of R and RS ÿ1 is a localized ring of R at l# . Let S be a set of L-singletons. De®ne the L-subset rS of R by 8x 2 R; rS …x† ˆ _fa j xa 2 Sg: If r 2 R and xa is an L-singleton, we let rxa denote the L-singleton …rx†a . Theorem 34. Let S be a set of L-singletons of R. Let r be the L-subset of R defined by 8x 2 R; ( ) ! k X ri …xi †ai …x† ri 2 R; xai 2 S; i ˆ 1; . . . ; k; k 2 N : r…x† ˆ _ iˆ1 Then r ˆ hSi; where hSi ˆ hrS i. Lemma 35. Suppose that l is an L-ideal of R such that l has the sup property. Let S ˆ [a2‰0;1Š Sa ; where Sa  fxa j x 2 R; l…x† ˆ ag if a 2 Im…l† and Sa ˆ ; if a 2 L n Im…l†. Then l ˆ hSi [ 0l…0† if and only if la ˆ …foot…[b P a Sb †† 8a 2 Im…l†. Proposition 36. Suppose l is an L-ideal of R such that l has the sup property. Let S ˆ [a2‰0;1Š Sa ; where Sa  fxa j x 2 R; l…x† ˆ ag if a 2 Im…l† and Sa ˆ ; if a 2 L n Im…l†. If foot…[b P a …Sb † is a minimal generating set for la 8a 2 Im…l†; then S is a minimal generating set for l. De®nition 37. Let S denote a set of L-singletons such that if xa and xb 2 S; then a ˆ b > 0. Let l be an L-ideal of R. Then S is called an S-minimal generating set for l if l ˆ hSi [ 0l…0† and 8x 2 foot…S†; there does not exist s 2 S such that sx 2 …foot…S† n fxg†. Proposition 38. Let S denote a set of L-singletons such that if xa and xb 2 S; then a ˆ b > 0. Let l be an L-ideal of R such that l has the sup property. If S is an S-minimal generating set for l; then S is a minimal generating set for l. If xa is an L-singleton of R; then /…xa † ˆ /…x†a . Let l and l0 be L-ideals of R and RS ÿ1 ; respectively, such that l0 is a localized L-subring of l in RS ÿ1 . If S is a set of L-singletons which generate l; then f/…x†a j xa 2 Sg generates l0 and we say that l and l0 have the same set of generators and we write /…S† for f/…x†a j xa 2 Sg.

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Theorem 39. Let k; k0 be L-ideals of R; RS ÿ1 ; respectively, such that k has the sup property. If k has an S-minimal generating set and k0 is a localized L-subring of k in RS ÿ1 ; then k and k0 have the same minimal generating sets and Im…k† ˆ Im…k0 †. Conversely, if RS ÿ1 is a localized ring at a prime ideal of R; k and k0 have the same minimal generating sets and Im…k† ˆ Im…k0 †; then k0 is a localized L-subring of k in RS ÿ1 . We now apply our results in the following example. Example 40. Let L ˆ ‰0; 1Š. Let R denote the polynomial ring R‰x; y; zŠ in the algebraically independent indeterminates x; y; z over the ®eld of R of real numbers. Then the ideal hx2 ÿ y; x2 zi represents the nonlinear system of equations x2 ÿ y ˆ 0;

x2 z ˆ 0

and has the reduced primary representation hx2 ÿ y; x2 zi ˆ hx2 ÿ y; zi \ hx2 ; yi: Hence p hx2 ÿ y; x2 zi ˆ hx2 ÿ y; zi \ hx; yi and the prime ideals hx2 ÿ y; zi and hx; yi display the solution of the nonlinear system of equations via their corresponding irreducible ane varieties. Now consider the following nonlinear system of fuzzy intersection equations …xb †2 ÿ ya ˆ 01=4 ;

…xb †2 zu ˆ 01=2 :

Then this system is represented by the fuzzy ideal l ˆ h…x2 ÿ y†1=4 ; …x2 z†1=2 i and S ˆ f…x2 ÿ y†1=4 ; …x2 z†1=2 g is a minimal generating set for l. In order to examine the system locally, we consider either of the prime ideals hx2 ÿ y; zi and hx; yi; say, P ˆ hx; yi; and we form the quotient ring RP . Then in RP ; the extended ideal [18] of hx2 ÿ y; zi is e

e

hx2 ÿ y; x2 zi ˆ hx2 ; yi : Hence the corresponding nonlinear system of fuzzy intersection equations is ya ˆ 01=4 ;

2

…xb † ˆ 01=2 :

This system is represented by the fuzzy ideal m ˆ hy1=4 ; …x2 †1=2 i in R. Now S ˆ fy1=4 ; …x2 †1=2 g is a minimal generating set for m. By Theorem 39, we have that S is a minimal generating set for the fuzzy localized subring m0 of m in RP . Hence m0 represents the same system of fuzzy intersection equations as m does. If we consider the prime ideal N ˆ hx2 ÿ y; zi; then in RN e

e

hx2 ÿ y; x2 zi ˆ hx2 ÿ y; zi :

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Hence the corresponding nonlinear system of fuzzy intersection equations is 2

…xb † ÿ ya ˆ 01=4 ;

zu ˆ 01=2 :

This system is represented by the fuzzy ideal k ˆ h…x2 ÿ y†1=4 ; z1=2 i in R. We have that f…x2 ÿ y†1=4 ; z1=2 g is a minimal generating set for k and also for the fuzzy localized subring k0 of k in RN . 5. Local examination of fuzzy intersection equations In this section, L denotes a dense chain and R a commutative ring with identity. The notion of algebraic L-varieties was introduced in order to use primary representation theory of L-ideals to examine the solution of fuzzy intersection equations. Local concepts of subrings were developed in order to lay the ground work for the examination of fuzzy intersection equations locally. In this section, we carry out a local examination of fuzzy intersection equations. We show that a system of fuzzy intersection equations can be examined locally to obtain the general solution to the crisp part of the system. The details can be found in [2]. Let M be a multiplicative system in R [18, p. 46]. Let N ˆ fx 2 R j mx ˆ 0 for some m 2 Mg. Then N is an ideal of R. If N ˆ f0g; then M is said to be regular. Let h be the natural homomorphism of R onto R=N  RM ; the quotient ring of R with respect to M. If I is an ideal of R; then the ideal in RM e generated by h…I† is called the extended ideal of I in RM and is denoted by h…I† . ÿ1 If J is an ideal of RM ; then h …J † is called the contracted ideal of J in R. e Let l be an L-ideal of R. De®ne the L-subset h…l† of RM by e e 8y 2 RM ; h…l† …y† ˆ _fa 2 L j y 2 …h…l††a Mg. Then h…l† is an L -ideal of RM . Let a 2 L. Now y 2 h…l†a () h…l†…y† P a () _ fl…x† j h…x† ˆ yg P a ( 9x 2 la such that h…x† ˆ y () y 2 h…la †; where the ``( '' becomes `` () '' if l has the sup property. Hence if l has the sup property, then h…l†a ˆ h…la † and so …h…l††aM ˆ h…la †M . We use the notation le for h…l†e at times. If I is an ideal e of R; we sometimes use the notation I e for h…I† . If b is an L-ideal of RM ; then c ÿ1 we use the notation b for h …b† at times. If J is an ideal of RM ; we sometimes use the notation J c for hÿ1 …J †. e Suppose that l has the sup property. Then …le †…y† ˆ a () h…l† …y† ˆ a () _ fb j y 2 h…l†bM ˆ h…lb †M g ˆ a () a is maximal in L such that y 2 h…l†aM ˆ h…la †M ˆ h…la †e …since l has the sup property† ˆ lea . Hence …le †a ˆ lea 8a 2 L. Theorem 41. Let b be a primary L-ideal of RM . Then

pc p c b ˆ … b† .

Theorem 42. Let l be a primary L-ideal of R such that l is disjoint from M. p p ec 1. Then l ˆ lec and l ˆ … l† . p  p e 2. Then le is primary and le ˆ … l† .

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c

Lemma 43. Let l and m be L-ideals of RM . Then …l \ m† ˆ lc \ mc . Theorem 44. Let l be an L-ideal of R such that l has a reduced primary representation l ˆ \niˆ1 li . Suppose that for 1 6 i 6 k; …li † \ M ˆ ; and that for k ‡ 1 6 i 6 n; …li † \ M 6ˆ ;. Then le ˆ \kiˆ1 lei is a reduced primary representation. Furthermore, lec ˆ \kiˆ1 li . Example 45. Let L ˆ ‰0; 1Š. Let R denote the polynomial ring R‰x; y; zŠ in algebraically independent indeterminates x; y; z over the ®eld R of real numbers. Then the ideal hx2 ÿ y; x2 zi has the reduced primary representation hx2 ÿ y; x2 zi ˆ hx2 ÿ y; zi \ hx2 ; yi: We also have p hx2 ÿ y; x2 zi ˆ hx2 ÿ y; zi \ hx; yi: Now consider the nonlinear system of fuzzy singletons 2

…xb † ÿ ya ˆ 01=4 ;

2

…xb † zu ˆ 01=2 :

…4†

The solution to system (4) is f…0; 0; r† j r 2 Rg [ f…s; s2 ; 0† j s 2 Rg; a ˆ 14; b ^ u ˆ 12. Let 8 1 if > > > < 1 if l…r† ˆ 21 > if > > :4 0 if

l denote the fuzzy ideal h…x2 ÿ y†1=4 ; …x2 z†1=2 i [ 01 . Then r ˆ 0; r 2 hx2 zi n f0g; r 2 hx2 ÿ y; x2 zi n hx2 zi; r 2 R n hx2 ÿ y; x2 zi:

De®ne the fuzzy subset li of R, i ˆ 1; . . . ; 5; as follows: l1 …r† ˆ 1 l2 …r† ˆ 1 l3 …r† ˆ 1 l4 …r† ˆ 1 l5 …r† ˆ 1

if if if if if

r 2 hx2 ; yi and 0 otherwise, r 2 hx2 ÿ y; zi and 0 otherwise, r 2 hx2 i and 14 otherwise, r 2 hzi and 14 otherwise, r 2 h0i and 12 otherwise.

Then li is a primary fuzzy ideal of R, i ˆ 1; . . . ; 5 and l ˆ \5iˆ1 li is a primary representation of l. Now p l …r† ˆ 1 if r 2 hx; yi and 0 otherwise, p1 l …r† ˆ 1 if r 2 hx2 ÿ y; zi and 0 otherwise, p2 l …r† ˆ 1 if r 2 hxi and 14 otherwise, p3 l …r† ˆ 1 if r 2 hzi and 14 otherwise, p4 l5 …r† ˆ 1 if r 2 h0i and 12 otherwise.

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Hence we see that l ˆ \5iˆ1 li is a reduced primary representation of l. We see that the crisp part of the solution to system (4) is displayed by p p p p p l1 \ l2 , while the fuzzy part is displayed by l3 \ l4 \ l5 . (In order to see this more clearly, one should consider the irreducible fuzzy algebraic vap rieties corresponding to the li . Then one would be concerned with c…14† ˆ 12 1 1 1 rather than 4 and c…2† ˆ 4 rather than 12.) Consider the quotient ring RP ; where P is the prime ideal hx; yi. Since e e P \ cP ˆ ;; we have in RP that hx2 ÿ y; x2 zi ˆ hx2 ; yi by [18, Theorem 17, p. 225]. Now l1 \ cP ˆ hx2 ; yi \ cP ˆ ;; l3 \ cP ˆ hx2 i \ cP ˆ ;; l5 \ cP ˆ h0i \ cP ˆ ;; while l2 \ cP ˆ hx2 ÿ y; zi \ cP 6ˆ ;; l4 \ cP ˆ hzi \ cP 6ˆ ;: Thus by Theorem 8 0 > > > <1 le …r† ˆ 41 > > > :2 1

44, we have in RP that le ˆ le1 \ le3 \ le5 and so if r 2 RP n hx2 ; yie ; e

e

if r 2 hx2 ; yi n hx2 i ; e

if r 2 hx2 i n f0g; if r 2 f0g:

Hence by Theorem 44,

lec …r† ˆ l1 \ l3 \ l5 …r† ˆ

8 0 > > > > <1 4 1 > > >2

> :

1

if r 2 R n hx2 ; yi; if r 2 hx2 ; yi n hx2 i; if r 2 hx2 i n f0g; if r 2 f0g:

Consider the nonlinear system of fuzzy singletons ya ˆ 01=4 ;

…xb †2 ˆ 01=2 :

…5†

Then f…0; 0; r† j r 2 Rg; a ˆ 14; b ˆ 12 is the solution to this system. It is represented by the fuzzy ideal m ˆ h…x1=2 †2 ; y1=4 i [ 01 . Now m ˆ l1 \ l3 \ l5 is a rep duced primary representation of m. l1 displays the crisp part of the solution p p while l3 \ l5 displays the fuzzy part. Now consider the prime ideal e N ˆ hx2 ÿ y; zi. Since N \ cN ˆ ;, we have in RN that hx2 ÿ y; x2 zi ˆ e hx2 ÿ y; zi . Now l2 \ cN ˆ hx2 ÿ y; x2 zi \ cN ˆ ;;

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l4 \ cN ˆ hzi \ cN ˆ ;; l5 \ cN ˆ h0i \ cN ˆ ;; while l1 \ cN ˆ hx2 ; yi \ cN 6ˆ ;; l3 \ cN ˆ hx2 i \ cN 6ˆ ;: Thus by Theorem 44, we have in RN that le ˆ le2 \ le4 \ le5 and so 8 e 0 if r 2 RN n hx2 ÿ y; zi ; > > >1 e < if r 2 hx2 ÿ y; zi n hzie ; l…r† ˆ 41 e > if r 2 hzi n f0g; > > :2 1 if r 2 f0g: Hence by Theorem 44,

lec …r† ˆ l2 \ l4 \ l5 …r† ˆ

8 0 > > >1 < 4 1 > > >2

:

1

if r 2 R n hx2 ÿ y; zi; if r 2 hx2 ÿ y; zi n hzi; if r 2 hzi n f0g; if r 2 f0g:

Consider the nonlinear system of fuzzy singletons 2

…xb † ÿ ya ˆ 01=4 ; 2

zu ˆ 01=2 : 1 ; 4

…6† 1 2

Then f…s; s ; 0† j s 2 Rg; b ^ a ˆ and u ˆ is the solution to this system. The system is represented by the fuzzy ideal k ˆ h…x2 ÿ y†1=4 ; z1=2 i [ 01 . Now p k ˆ l2 \ l4 \ l5 is a reduced primary representation of k. l2 displays the p p crisp part of the solution, while l4 \ l5 displays the fuzzy part. We have examined the system (4) locally. From the two examinations, we obtain for the crisp part of the solution f…0; 0; r† j r 2 Rg for (5) and f…s; s2 ; 0† j s 2 Rg for (6). The union of these two gives us the crisp part of the solution to system (4). However, the fuzzy solutions to (5) and (6) are a ˆ 14; b ˆ 12 and b ^ a ˆ 14; u ˆ 12; respectively. The fuzzy part of the solution to (4) is a ˆ 14 and b ^ u ˆ 12. The two ``local'' fuzzy solutions do not seem to give us the fuzzy part of the solution to (4), at least not immediately. Consider all possible ^'s of the two fuzzy solutions above a ^ b ^ a ˆ 14 ^ 14; a ^ u ˆ 14 ^ 12;

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b ^ b ^ a ˆ 12 ^ 14; b ^ u ˆ 12 ^ 12: These equations reduce to b ^ a ˆ 14; a ^ u ˆ 14; b ^ a ˆ 14; b ^ u ˆ 12: Hence a ˆ 14 and b ^ u ˆ 12 which is the solution to the original problem. Problem: Determine a general procedure to ®nd the solution to the fuzzy part of the original problem from the local solutions. An algorithm for solving fuzzy systems of intersection equations is given in [15] and an application to fuzzy graph theory is given in [16]. For a study if L-intersection equations for L a complete distributive lattice, the reader is referred to [5]. References [1] Y. Alkhamees, J.N. Mordeson, Fuzzy localized subrings, Inform. Sci. 99 (1997) 183±193. [2] Y. Alkhamees, J.N. Mordeson, Local examination of fuzzy intersection equations, FSS 98 (1998) 249±254. [3] N. Bourbaki, Elements of Mathematics, Commutative algebra, Springer, New York, 1989 (Chapters 1±7). [4] E. Eslami, J.N. Mordeson, Completions and fuzzy power series subrings, FSS 82 (1996) 97±102. [5] L. Wangjin, On some systems of simultaneous equations in a completely distributive lattice, Inform. Sci. 50 (1990) 185±196. [6] D.S. Malik, J.N. Mordeson, Fuzzy primary representations of fuzzy ideals, Inform. Sci. 55 (1991) 151±165. [7] D.S. Malik, J.N. Mordeson, Extensions of fuzzy subrings and fuzzy ideals, FSS 45 (1992) 245± 251. [8] D.S. Malik, J.N. Mordeson, R-primary L-representations of L-ideals, Inform. Sci. 88 (1996) 227±246. [9] N.H. McCoy, Rings and Ideals, The Carus Mathematical Monograms, No. 8, The Mathematical Association of America, 1956. [10] J.N. Mordeson, Fuzzy algebraic varieties, Rocky Mount. J. Math. 23 (1993) 1361±1377. [11] J.N. Mordeson, Fuzzy algebraic varieties II, in: P. Wang (Ed.), Advances in Fuzzy Theory and Technology, vol. I, 1993, pp. 9±21.

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[12] J.N. Mordeson, Fuzzy coecient ®elds of fuzzy subrings, FSS 58 (1993) 227±237. [13] J.N. Mordeson, Fuzzy intersection equations and primary representations, FSS 83 (1996) 93± 98. [14] J.N. Mordeson, D.S. Malik, Fuzzy Commutative Algebra, World Scienti®c, Singapore, 1998. [15] J.N. Mordeson, P. Chang-Shyh, Fuzzy intersection equations, FSS 60 (1993) 77±81. [16] J.N. Mordeson, P. Chang-Shyh, Operations on fuzzy graphs, Inform. Sci. 79 (1994) 159±170. [17] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512±517. [18] O. Zariski, P. Samuel, Commutative Algebra, vol. I, D. Van Nostrand, Princeton, NJ, 1958.