Fuzzy programming approach to multi-level programming problems

Fuzzy Sets and Systems 136 (2003) 105 – 114 www.elsevier.com/locate/fss Fuzzy star-operations on an integral domain  Hwankoo Kim ∗ , Myeong Og Kim, Sung-Mi Park, Young Soo Park Department of Mathematics, Kyungpook National University, Taegu 702-701, South Korea Received 23 February 2001; received in revised form 4 April 2002; accepted 22 May 2002 Abstract In this paper, we introduce the concept of fuzzy star-operations on an integral domain and show that the set of all fuzzy star-operations on the integral domain forms a complete lattice. We also characterize Prufer domains, psuedo-Dedekind domains, (generalized-) greatest common divisor domains, and other integral domains in terms of the invertibility of certain fractionary fuzzy ideals. c 2002 Elsevier Science B.V. All rights reserved.
Keywords: (Fuzzy)Algebra; Fractionary fuzzy ideal; Fuzzy invertible; Fuzzy star-operation; Prufer domain; PID; Pseudo-Dedekind domain; (G-)GCD domain; Pseudo-principal domain 1. Introduction Many papers on fuzzy algebras have been published since Rosenfeld [10] introduced the concept of a fuzzy subgroup in 1971. Recently, Lee and Mordeson [8] introduced the notion of fractionary fuzzy ideals and of fuzzy invertible fractionary fuzzy ideals, and using these notions they characterized Dedekind domains in terms of the invertibility of certain fractionary fuzzy ideals. To the best of our knowledge, this is the rst attempt of the fuzzi cation of one of the main results in multiplicative ideal theory. As in [4,5], star-operations are essential tools in the study of multiplicative ideal theory. In fact, the (ordinary) ideals of an integral domain are derived from trivial star-operation (i.e., the identity mapping from the set of all fractionary ideals of the integral domain into itself). Thus the class of Dedekind domains, which is the main subject of [7,8], is related to the trivial star-operation. For another example, there is the important class of Krull domains, whose properties are similar to  This work was supported by Grant No. R02-2001-00038 from the Basic Research Program of the Korea Science & Engineering Foundation. ∗ Corresponding author. Present address: Department of Information Security, Division of Computer Science, Hoseo University, Asan 336-795, South Korea. E-mail addresses: hkkim@oce.hoseo.ac.kr (H. Kim), yngspark@knu.ac.kr (Y.S. Park). c 2002 Elsevier Science B.V. All rights reserved. 0165-0114/03/$ - see front matter  PII: S 0 1 6 5 - 0 1 1 4 ( 0 2 ) 0 0 2 7 3 - 7 106 H. Kim et al. / Fuzzy Sets and Systems 136 (2003) 105 – 114 those of Dedekind domains. These Krull domains are also related to a star-operation, the so-called v-operation. Therefore, the investigation of fuzzy star-operations is the start of the study of fuzzy multiplicative ideal theory, more generally, fuzzy ideal systems. In this paper, we provide the foundation for the study of the fuzzy ideal systems. More precisely, in Section 2, we state de nitions and results from [7,8] for easy reference. In Section 3, we investigate fractionary fuzzy ideals, which will be used in later sections. In Section 4, we introduce the concept of fuzzy star-operations on an integral domain R. In particular, we de ne the fuzzy v-operation on an integral domain R. As one of the main results, we show that the set of all fuzzy star-operations on an integral domain R forms a complete lattice. In Section 5, we characterize Prufer domains, pseudo-Dedekind domains, pseudo-principal domains, principal ideal domains (PIDs), and (G-)GCD domains in terms of the invertibility of certain fractionary fuzzy ideals. In the nal section, we give a conclusion to this paper and give a question for further study. 2. Preliminaries Throughout this paper, we let R be an integral domain with quotient eld K. A fuzzy subset of R is a function from R into [0; 1]. Let ,  be fuzzy subsets of R. We write  ⊆ if (x)6(x) for all x ∈R. If  ⊆ and there exists x ∈R such that (x)¡(x), then we write  ⊂. We denote the image of  by Im(). We say that  is nite-valued if |Im()|¡∞. Let t ={x ∈R | (x)¿t}, a level set, for every t ∈[0; 1]. We let W denote the characteristic function of a subset W of R and let R(t) be the fuzzy subset of K such that R(t) (x)=1 if x ∈R and R(t) (x)=t if x ∈K − R, where t ∈[0; 1). A fuzzy subset  of R is a fuzzy ideal of R if for every x; y ∈R, (x − y)¿(x)∧(y) and (xy)¿(x)∨(y). A fuzzy subset  of R is a fuzzy ideal of R if and only if (0)¿(x) for every x ∈R and t is an ideal of R for every t ∈[0; (0)] A fuzzy subset of K is a fuzzy R-submodule of K if (x − y)¿ (x)∧ (y), (rx)¿ (x) and (0)=1 for every x; y ∈K, r ∈R. A fuzzy subset of K is a fuzzy R-submodule of K if and only if (0)=1 and t is an R-submodule of K for every t ∈[0; 1]. We let ∗ denote {x ∈K | (x)= (0)}. Let N denote the positive integers. Let and
be fuzzy subsets of K. De ne the fuzzy subset ◦ of K by: for every x ∈K, ( ◦ )(x)= { (y)∧ (z) | y; z ∈K} if x is expressible as a product x =yz for some y; z ∈K and
 n ( ◦ )(x)=0 otherwise. The product of and , written , is de ned by (x)= { i=1 ( (yi )∧ n (zi )) | yi ; zi ∈K; 16i6n; n∈N; i=1 yi zi =x}. Let { i | i =1; : : : ; n} be a collection of fuzzy subsets of n
 n K: we de ne the fuzzy subset of K by, for very x ∈K, ( )(x)= { { (x ) i=1 i  i=1 n  i i | i =1; : : : ;  i n} | x = i=1 xi ; xi ∈K}. A fuzzy subset i∈I i of K is de ned by ( i∈I i )(x)= { i (x) | i ∈I } for every x ∈K. For d∈K and t ∈[0; 1], we let dt denote the fuzzy subset of K de ned by: for every x ∈K, dt (x)=t if x =d and dt (x)=0 otherwise. We call dt a fuzzy singleton. Let  be a fuzzy subset of K and let  denote the intersection of all the fuzzy submodules of K which contain . Then  is called the fuzzy submodule of K generated by . From [7], we recall the following two de nitions, which are the fuzzi cation of important concepts in multiplicative ideal theory. • A fuzzy R-submodule of K is called a fractionary fuzzy ideal of R if there exists d∈R; d = 0, such that d1 ◦ ⊆R(t) for some t ∈[0; 1). 107 H. Kim et al. / Fuzzy Sets and Systems 136 (2003) 105 – 114 • A fractionary fuzzy ideal is said to be fuzzy invertible if there exists a fractionary fuzzy ideal ′ of R such that ′ =(t) for some t ∈[0; 1). R Any unexplained notation or terminology is standard as in [9]. For the purpose of easy reference, we include the following three results, which are essential for our study. For details, we refer the reader to [7,8]. Proposition 2.1 (Lee and Mordeson [7, Proposition 3.3]). Let be a fractionary fuzzy ideal of R, 0 = d∈R, and s∈[0; 1). Then d t ⊆R for all t ∈(s; 1] if and only if d1 ◦ ⊆R(s) . Theorem 2.2 (Lee and Mordeson [7, Theorem 4.2]). Let be a fractionary fuzzy ideal of R. Suppose that max{ (x) | x ∈K\ ∗ } exists. Then is fuzzy invertible if and only if ∗ is an invertible fractionary ideal of R. Theorem 2.3 (Lee and Mordeson [8, Theorem 2.3]). The following conditions on an integral domain R are equivalent. (i) R is a Dedekind domain. (ii) Every integral fractionary fuzzy ideal of R such that max{ (x) | x ∈K\ ∗ } exists is fuzzy invertible. (iii) Every fractionary fuzzy ideal of R such that max{ (x) | x ∈K\ ∗ } exists is fuzzy invertible. (iv) Every nite-valued fuzzy ideal  of R with (0)=1 is uniquely expressible as a product of a nite number of maximal fuzzy ideals. 3. Fractionary fuzzy ideals In this section, we study the operations from the set of fractionary fuzzy ideals into itself. The results obtained in this section will be used in the remaining sections. Lemma 3.1. Let { i | i ∈I } be a collection of fuzzy R-submodules of K and let 0 = d∈K. Then   (1) d1 ◦(i i )= i (d1 ◦ i ). (2) d1 ◦( i i )= i (d1 ◦ i ). Proof. Let x ∈K. Then (d1 ◦( Lemma 3.2. Let , ideals of R. Proof. Since d1 ◦ ⊆R(s) and (dd′ )1 ◦ ⊆R(t) . =(d1 ◦ )◦(d1′ ◦  i i ))(x)=  i x i ( d )=  i (d1 ◦ i )(x)=( be fractionary fuzzy ideals of R. Then +  i (d1 ◦ i ))(x). and ◦ are fractionary fuzzy and are fractionary fuzzy ideals of R, there exists 0 = d; 0 = d′ ∈R such that ′ d1 ◦ ⊆R(t) for some s; t ∈[0; 1). So (d′ d)1 ◦ =d1′ ◦d1 ◦ ⊆d1′ ◦R(s) ⊆R(s) . Similarly, Hence (dd′ )1 ◦( + )=(dd′ )1 ◦ + (dd′ )1 ◦ ⊆R(s) + R(t) ⊆R(t ∨s) and (dd′ )1 ◦( ◦ ) )⊆R(s) ◦R(t) ⊆R(t ∨s) . Therefore, + and ◦ are fractionary fuzzy ideals of R. 108 H. Kim et al. / Fuzzy Sets and Systems 136 (2003) 105 – 114 Proposition 3.3. Let 0 = xi ∈K and ai ∈[0; 1]; i =1; : : : ; n. Then (x1 )a1 ; : : : ; (xn )an = n i=1 (xi )ai . n )ai ⊆ (x1 )a1 ; : : : ; (x
Proof. For each i, (xi n )an ; thus i=1 (xi )ai ⊆  (x1 )a1 ; : : : ; (xn )an . The reverse  inclusion holds since ( ni=1 (xi )ai )(xj )= (a1 ∧ · · · ∧an ) if xj = ni=1 ri xi for some ri ∈R, ( ni=1 (xi )ai )(xj )=0, otherwise and (x1 )a1 ; : : : ; (xn )an = , where  is the fuzzy subset of K such that (xj )=a1 ∧ · · · ∧an for j =1; : : : ; n and (k)=0 if k = xj , j =1; : : : ; n. Corollary 3.4. The nite sum of nitely generated fractionary fuzzy ideals of R is a nitely generated fractionary fuzzy ideal of R. The following result is easy, but it is needed in the characterization of PIDs. Proposition 3.5. Every proper level set of a fuzzy principal ideal of R is a principal ideal. Proof. For 0 = x ∈K and a∈[0; 1]; xa t =(x) if 0 t6a and xa t =(0) otherwise. Corollary 3.6. Every proper level set of a nitely generated fractionary fuzzy ideal of R is nitely generated. Proof. Let  = (x1 )a1 ; : : : ; (xn )an  be any nitely generated fractionary fuzzy ideal of R and let t∈[0; 1]. Then t = (x1 )a1 ; : : : ; (xn )an t = (x1 )a1 t +· · ·+ (xn )an t =(x1 )+· · ·+(xn ) if t∈(0; ai ]; i=1; : : : ; n and t =(0) otherwise. Let and be fuzzy R-submodules and x ∈K. We recall from [11] that the residual quotient
( : ) de ned by ( : )(x)= {l∈[0; 1] | xl ◦ ⊆ } is a fuzzy R-submodule. Lemma 3.7. Let be a nitely generated fractionary fuzzy ideal of R with d1 ◦ ⊆R(s1 ) and a fractionary fuzzy ideal of R with d1′ ◦ ⊆R(s2 ) , where 0= d1 ; 0= d2 ∈R, s1 ; s2 ∈[0; 1). Then ( : )t ⊆( t : t ) for every t ∈(s; 1], where s=s1 ∨s2 . Proof. (i) Let = xa  for some 0 = x ∈K and a∈[0; 1]. If t a, then ( t : t )=0. We show that ( : )t =0. If
not, there exists 0 = y ∈( : )t and yw = 0 for some w ∈ t . Let L={l∈[0; 1] | yl ◦ ⊆ }. Since L¿t 0, we have L = {0}. Then (yl ◦ )(yw)6 xa (yw) for every 0 = l∈L. Thus l∧ (w)6a if yw ∈(x) and l∧ (w)=0 otherwise, which is a contradiction. If t6a, then t =(x) by Proposition 3.5. Let y ∈( : )t and suppose that y t *(x). Then there exists w ∈ t such that yw ∈(x). = Since L={l∈[0; 1] | yl ◦ ⊆ } =  {0}, we have l∧ (w)=(yl ◦ )(yw)6 xa (yw)=0 for every 0 = l∈L. This is a contradiction. (ii) Let = (x1 )a1 ; : : : ; (xn )an  for some a1 · · · an ; n¿2. In this case, the proof is similar to (i). Remark 3.8. In Lemma 3.7, the reverse containment does not hold in general. For example, let Q be the eld of rational numbers and Z the ring of integers. De ne the fuzzy subset of Q by (x)=1 if x ∈2Z, (x)= 12 if x ∈Z\2Z, and (x)=0 if x ∈Q\Z. Then is a fuzzy Z-submodule of Q. Since (1)1 ◦ ⊆Z(0) , is a fractionary fuzzy ideal of Z. Now since 1∈(Z(0) : )1=2 , we have (Z(0) : )1=2 =Z=(Z: 1=2 ). But (Z: 2=3 )= 12 Z*(Z(0) : )2=3 since 12 ∈( = Z(0) : )2=3 . 109 H. Kim et al. / Fuzzy Sets and Systems 136 (2003) 105 – 114 Theorem 3.9. Let be a nitely generated fractionary fuzzy ideal of R and ideal of R. Then ( : ) is a fractionary fuzzy ideal of R. a fractionary fuzzy Proof. Let 0 = d; 0 = d′ ∈R such that d1 ◦ ⊆R(s1 ) and d1′ ◦ ⊆R(s2 ) where s1 ; s2 ∈[0; 1). Let t ∈(s; 1], where s=s1 ∨s2 . Then for any 0 = b∈ t , bd( : )t ⊆bd( t : t )⊆d t ⊆R by Lemma 3.7. Thus (bd)1 ◦ ( : )⊆R(s) by Proposition 2.1. Therefore ( : ) is a fractionary fuzzy ideal of R. Corollary 3.10. Let of R. be a fractionary fuzzy ideal of R. Then (R(0) : ) is a fractionary fuzzy ideal 4. Fuzzy star-operations In this section, we de ne a fuzzy star-operation on R and show that the set of all fuzzy staroperations forms a complete lattice. Denote the set of all fractionary fuzzy ideals of R by Fz (R). De nition 4.1. A fuzzy star-operation on R is a mapping → three properties for all 0 = d∈K and ; ∈Fz (R): (1) (R(s) )∗ =R(s) and (d1 ◦ )∗ =d1 ◦ (2) ⊆ ∗ ; if ⊆ , then ∗ ⊆ ∗ . (3) ( ∗ )∗ = ∗ . It is clear that ⊆ ∗. ∗ on Fz (R) that satis es the following ∗. As in multiplicative ideal theory, we de ne a fuzzy ∗-ideal as follows: De nition 4.2. A fractionary fuzzy ideal is called a fuzzy ∗-ideal if ∗= . We will denote the set of all fuzzy ∗-ideals of R by Fz∗ (R). The set of all fuzzy ∗-operations on R will be denoted by z (R). Proposition 4.3. Let { i | i ∈I } be a collection of fractionary fuzzy ideals of R and ∗∈z (R). Then   =( i ∗i )∗ (1) ( i i )∗  (2) i ∗i =( i ∗i )∗ . if (  i ∗ ∈F (R). i) z   ∗    ∗ ∗ Proof. (1) For each i, i ⊆ i i ; thus ∗i ⊆( i i )∗ . Therefore i ) . Since ( i ⊆( i i) i i    ∈Fz (R), there exists 0 = d∈R such that d1 ◦( i i )∗ ⊆R(t) for some t. So d1 ◦ i ∗i ⊆d1 ◦( i i )∗ ⊆   ∗ ∗   ∗ ∗ ∗∗ R(t) . Hence i i ) . The reverse containment is clear i i ) =( i i ) ⊆( i i ∈Fz (R) and (   ∗ since i i ⊆ i i .  ∗  ∗ ∗  ∗ ∗ (2) We need only show that ( . But this follows since the containment ) ⊆ i i i i ⊆ j i i  ∗ ∗ for each j implies that ( i i ) ⊆ j∗∗ = j∗ for each j. The following result is easy to prove, but it is essential in our study. 110 H. Kim et al. / Fuzzy Sets and Systems 136 (2003) 105 – 114 Lemma 4.4. Let be a fractionary fuzzy ideal of R. De ne  {x1 ◦ R(s) | ⊆ x1 ◦ R(s) ; 0 = x ∈ K and s ∈ [0; 1)}: v = Then v is a fractionary fuzzy ideal of R containing . In multiplicative ideal theory, many integral domains are related to the v-operation. Thus, the following result provides useful tools to study fuzzy multiplicative ideal theory. Theorem 4.5. Let be a fractionary fuzzy ideal of R. Then the mapping fuzzy star-operation on R, called the fuzzy v-operation on R. → v on Fz (R) is a Proof. Since R(s) =(1)1 ◦R(s) , we have R(s) =(R(s) )v . If y1 ◦R(t) is any fractionary fuzzy ideal of R  containing , then d1 ◦ ⊆d1 ◦(y1 ◦R(t) ), and so (d1 ◦ )v ⊆d1 ◦(y1 ◦R(t) ). Thus (d1 ◦ )v ⊆ {d1 ◦(y1 ◦  R(t) ) | ⊆y1 ◦R(t) }=d1 ◦ {y1 ◦R(t) | ⊆y1 ◦R(t) }=d1 ◦ v . And if x1 ◦R(s) is any fractionary fuzzy ideal of R containing d1 ◦ , then ⊆(x=d)1 ◦R(s) . Hence d1 ◦ v ⊆x1 ◦R(s) , and so d1 ◦ v ⊆(d1 ◦ )v . Therefore condition (1) of the de nition holds. It is clear from the de nition of the mapping that condition (2) holds. If y1 ◦R(t) is any fractionary fuzzy ideal of R containing , then v ⊆(y1 ◦R(t) )v =y1 ◦ (R(t) )v =y1 ◦R(t) . Thus ( v )v ⊆ v , and so v =( v )v . Therefore, condition (3) of the de nition holds. By De nition 4.2 and Theorem 4.5, a fractionary fuzzy ideal = . v of R is called a fuzzy v-ideal if Proposition 4.6. Let be a fractionary fuzzy ideal of R with d1 ◦ ⊆R(s) , where 0 = d∈R and s∈[0; 1). If is a fuzzy v-ideal, then t is a v-ideal for any t ∈(s; 1]. Proof. Let be a fuzzy v-ideal of R and t ∈(s; 1]. Then ′ ′ ′   {y1 ◦R(s ) | ⊆y1 ◦R(s ) })t = {(y1 ◦R(s ) )t | t =( v )t =( Hence ( t )v ⊆ t ⊆( t )v . Thus ( t )v = t .  (s′ ) {(y) | t ⊆(y1 ◦R )t }= t ⊆(y)}. The following result is the fuzzi cation of a trivial result in multiplicative ideal theory. Proposition 4.7. A fractionary fuzzy principal ideal of R is a fuzzy v-ideal. Proof. Let 0 = x ∈K and a∈[0; 1]. Then ( xa )v =( and s ′ ∈[a; 1)})= xa .  {x1 ◦R(s) | 06s 1})∩(  ′ {y1 ◦R(s ) | y = x In multiplicative ideal theory, it is always true that Iv =R:(R:I ) for any nonzero ideal I of an integral domain R, where Iv is de ned as the intersection of all principal fractional ideals of R containing I . Does the converse of the following result hold? Proposition 4.8. Let be a fractionary fuzzy ideal of R. Then (0) (0) v ⊆R :(R : ) H. Kim et al. / Fuzzy Sets and Systems 136 (2003) 105 – 114 111   (s)  (s) (s) (s) Proof.  Let 0 = w ∈K. Then v (w)= {x1 ◦R (w) | ⊆x1 ◦R }= {R (w=x) | ⊆x1 ◦R }= s. Set s=t. Clearly the result is true for t =0. Assume that t = 0 and we claim that t ∈{l∈[0; 1] | wl ◦ (R(0) : )⊆R(0) }. Let y ∈R. = Then wt ◦(R(0) : )(y)=t ∧(R(0) : )(z), where y =wz. Suppose that (R(0) : )(z)= 0. Then there exists 0 = m such that (zm ◦ )(y)6R(0) (y). So m∧ (w)=0. Hence (w)=0 for any 0 = w ∈K. Thus (w)=0 if w = 0 and (w)=1 if w =0. Therefore t = v (w)6R(0) (w). So t =0. This is a contradiction. Thus wt ◦(R(0) : )⊆R(0) . We may de ne a partial order 6 on z (R) by ∗1 6∗2 if and only if ∗1 ⊆ ∗2 for every ∈Fz (R). For ∗1 ; ∗2 ∈z (R), it is easily seen that the following conditions are equivalent: (1) (2) (3) (4) ∗1 6∗2 . ( ∗2 )∗1 = ∗2 for every ( ∗1 )∗2 = ∗2 for every Fz∗2 (R)⊆Fz∗1 (R). ∈Fz (R). ∈Fz (R). Note that under this partial order, there is a smallest element, the fuzzy d-operation ( d = for every ∈Fz (R)), and a greatest element, the fuzzy v-operation. The proof of the following result is routine and so we omit it. Lemma 4.9. Suppose that  {∗i } is a nonempty collection of fuzzy star-operations on R. For each ∈Fz (R), de ne ∗ by ∗ = i ∗i . Then ∗ is a fuzzy star-operation on R. Suppose that {∗ i } is a nonempty collection of fuzzy star-operations on R. For each ∈Fz (R), de ne ∗ by ∗ = i ∗i . Then by Lemma 4.9, ∗ is a fuzzy star-operation on R. Clearly∗6∗i for ′ each i. If ∗′ is any other fuzzy star-operation on R with ∗′ 6∗i for each i, then ∗ ⊆ i ∗i = ∗ . Hence ∗′ 6∗. Thus ∗ is the meet of the collection {∗i }. Since z (R) is a partially ordered set with the greatest is closed under arbitrary meets, z (R) is a complete lattice with join
element  v and z given by ∗i = {∗∈ (R) |∗¿∗i for each i}. We record this fact, the fuzzi cation of the main result in [2], as Theorem 4.10. z (R) is a complete lattice with respect to the partial order 6. 5. Characterizations of some integral domains We recall that a fractionary fuzzy ideal is said to be fuzzy invertible if there exists a fractionary fuzzy ideal ′ of R such that ′ =R(t) for some t ∈[0; 1). If there exists a fractionary fuzzy ideal ′ of R such that ′ =R(0) , then is said to be invertible. Note that x1  is invertible, but xa  is not fuzzy invertible if a = 1. As in Theorem 2.3, Lee and Mordeson characterized Dedekind domains in terms of fuzzy invertibility. In this section, we characterize several integral domains in terms of fuzzy invertibility. We start with an easy observation, and so we omit its proof. Proposition 5.1. (1) Let x ∈K\{0}. Then the fuzzy subset x of K de ned by x (k)=1 if k =0, x (k)=a (a∈[0; 1)) if k ∈(x)\{0} and x (k)=0 otherwise, is a fuzzy R-submodule of K and x = xa  112 H. Kim et al. / Fuzzy Sets and Systems 136 (2003) 105 – 114 (2) There is a one-to-one correspondence between the set  of fuzzy principal ideals x1  of R and the set of nonzero principal ideals of R given by x1  →  x1 ∗ for all x ∈K\{0}. An integral domain R is called a Prufer domain if every nonzero nitely generated ideal of R is invertible. Clearly a Dedekind domain is a Prufer domain. For any unexplained notation or terminology on integral domains, one may consult [4,6]. Theorem 5.2. The following conditions are equivalent for an integral domainR. (1) R is a Prufer domain. (2) Every nitely generated integral fractionary fuzzy ideal of R such that ∗ = 0 is fuzzy invertible. (3) Every nitely generated fractionary fuzzy ideal of R such that ∗ = 0 is fuzzy invertible. Proof. (1)⇒(3): Let = (x1 )a1  + · · · + (xn )an  be a nitely generated fractionary fuzzy ideal of R such that ∗ = 0. Since ∗ = 0, we have each ai =1. By Corollary 3.6, ∗ =(x1 ) + · · · + (xn ). Hence is fuzzy invertible. ∗ is invertible. By Theorem 2.2, (3)⇒(2): This is trivial. (2)⇒(1): Let I =(x1 ; : : : ; xn ) be a nitely generated ideal of R. Then = (x1 )1  + · · · + (xn )1  is a nitely generated integral fractionary fuzzy ideal of R such that ∗ = 0 by Proposition 5.1. Hence is fuzzy invertible by assumption. Again by Theorem 2.2, ∗ =(x1 ) + · · · + (xn )=I is invertible. Therefore R is a Prufer domain. Let R be an integral domain with quotient eld K. If I is an integral ideal of R, then any subset of K of the form (1=d)I , where d is a nonzero element of R, is called a fractional ideal of R. For a fractional ideal I of an integral domain R with quotient eld K, Iv is de ned as the fractional ideal R:(R:I )=(I −1 )−1 . A fractional ideal I is called a v-ideal if Iv =I . Recall from [3] that an integral domain R is called a pseudo-Dedekind (resp., pseudo-principal) domain if every v-ideal of R is invertible (resp., principal). Theorem 5.3. R is a pseudo-Dedekind domain if and only if every fuzzy v-ideal max{ (x) | x ∈K\ ∗ } exists is fuzzy invertible. of R such that Proof. Let R be a pseudo-Dedekind domain and let be a fuzzy v-ideal of R such that max{ (x) | x ∈ K\ ∗ } exists. Then ∗ is a v-ideal of R by Proposition 4.6. Since R is a pseudo-Dedekind domain, ∗ is an invertible ideal of R. By Theorem 2.2, is fuzzy invertible. Conversely, let I be a v-ideal of R and let {() }∈ be the family of principal fractionary ideals of R which contain  I . Then { ( )1 }∈ is a family of fuzzy principal ideals of R such that ( )1 ∗ =( ). Let =  { ( )1  | I ⊆( )}. Then is a fuzzy v-ideal of R such that max{ (x) | x ∈K\ ∗ } exists. Hence is a fuzzy invertible fractionary fuzzy ideal of R by assumption. Again by Theorem 2.2, ∗ =I is an invertible ideal of R. Therefore R is a pseudo-Dedekind domain. Theorem 5.4. R is a pseudo-principal domain if and only if every fuzzy v-ideal ∗ = 0 is a fuzzy principal ideal of R. of R such that 113 H. Kim et al. / Fuzzy Sets and Systems 136 (2003) 105 – 114 Proof. Let R be a pseudo-principal domain and let be a fuzzy v-ideal of R. Then ∗ is a v-ideal of R. Since R is a pseudo-principal domain, ∗ is a principal ideal of R, say ∗ =(x) for some x ∈K\{0}. Note that x1 ⊆ . We will show that = x1 . Suppose that is not a fuzzy principal ideal of R. Then there exists 0 = y ∈K such that x1 ; ya ⊆ and x1 ; ya  =  x1 , where a∈(0:1]. Since 0 = ∗ =(x)⊆( x1  + ya )∗ , we have a=1. Then ( x1  + y1 )∗ ⊆ ∗ =(x). Hence (x; y)=(x), and so x1 ; y1 = x1 . This is a contradiction. The proof of reverse implication is similar to that of Theorem 5.3. An integral domain R is called a PID if every ideal of R is principal. It is clear that a PID is a Dedekind domain. Theorem 5.5. R is a PID if and only if every fractionary fuzzy ideal fuzzy principal ideal of R. of R such that ∗ = 0 is a Proof. Suppose that every fractionary fuzzy ideal of R such that ∗ = 0 is fuzzy principal ideal of R, and let I be an ideal of R. Then I is a fractionary fuzzy ideal of R such that ∗ = 0. Hence I is a fuzzy principal ideal of R. Thus I =(I )∗ is a principal ideal of R. Therefore R is a PID. The proof of reverse implication is similar to that of Theorem 5.4. Recall from [1] that an integral domain R is called a GCD (resp., G-GCD) domain if the intersection of every two principal ideals of R is principal (resp., invertible). Theorem 5.6. R is a GCD domain if and only if the intersection of two fuzzy principal ideals and such that ( ∩ )∗ = 0 is a fuzzy principal ideal of R. Proof. Let R be a GCD domain and let and be fuzzy principal ideals of R such that ( ∩ )∗ = 0. Then = x1  and = y1  for some x; y ∈K\{0}. Hence ∗ =(x) and ∗ =(y) by Proposition 3.5. Since R is a GCD domain, ( ∩ )∗ = ∗ ∩ ∗ is a principal ideal of R, say, ( ∩ )∗ =(z). Hence ∩ = z1  is a fuzzy principal ideal of R. Conversely, let A=(x) and B=(y) be principal ideals of R. Then x1  and y1  are fuzzy principal ideals of R such that ( x1 ∩ y1 )∗ = 0. By assumption, x1 ∩ y1  is a fuzzy principal ideal of R, say, za  for some z ∈K\{0} and a∈[0; 1]. Since ( x1 ∩ y1 )∗ = 0, we have a=1. Hence A∩B=(x)∩(y)= x1 ∗ ∩ y1 ∗ = z1 ∗ =(z) is a principal ideal of R. Therefore R is a GCD domain. Theorem 5.7. R is a G-GCD domain if and only if for all fuzzy invertible fractionary fuzzy ideals and of R such that max{ (x) | x ∈K\ ∗ } and max{ (x) | x ∈K\ ∗ } exist, ∩ is fuzzy invertible. Proof. Let R be a G-GCD domain, and let and be fuzzy invertible fractionary fuzzy ideals of R such that max{ (x) | x ∈K\ ∗ } and max{ (x) | x ∈K\ ∗ } exist. Then ∗ and ∗ are invertible ideals of R. Since R is a G-GCD domain, ( ∩ )∗ = ∗ ∩ ∗ is an invertible ideal of R. Since max{( ∩ )(x) | x ∈K\( ∩ )∗ } exists, ∩ is fuzzy invertible by Theorem 2.2. Conversely, let A and B be invertible ideals of R. Then  A and  B are fuzzy invertible fractionary fuzzy ideals of R such that max{ A (x) | x ∈K\( A )∗ } and max{ B (x) | x ∈K\( B )∗ } exist. Thus  A ∩ B is a fuzzy 114 H. Kim et al. / Fuzzy Sets and Systems 136 (2003) 105 – 114 invertible fractionary fuzzy ideal of R such that max{( A ∩ B )(x) | x ∈K\( A ∩ B )∗ } exists. Hence A∩B=( A )∗ ∩( B )∗ =( A ∩ B )∗ is an invertible ideal of R. Therefore R is a G-GCD domain. 6. Conclusion In multiplicative ideal theory, star-operations on integral domains play an important role in the study of the integral domains. It is well known that the set of all star-operations on an integral domain forms a complete lattice [2]. In this paper, we introduced the new concepts of fuzzy star-operations and fuzzy star ideals, and fuzzi ed the above result. Using these concepts, we characterized several integral domains including Prufer domains and PIDs. The results achieved in this paper, together with the results in [7,8], will provide a solid foundation for studying fuzzy multiplicative ideal theory. The class of Krull domains is one of the main classes in multiplicative ideal theory, and their properties are very similar to those of Dedekind domains. Thus it is natural to ask whether there exists a similar result as in Theorem 2.3. Acknowledgements The authors take this opportunity to thank the referees for their helpful comments. References [1] D.D. Anderson, D.F. Anderson, Generalized GCD domains, Comment Math. Univ. St. Pauli XXVIII-2 (1979) 215–221. [2] D.D. Anderson, D.F. Anderson, Examples of star-operations on integral domains, Comm. Algebra 18 (1990) 1621–1643. [3] D.D. Anderson, B.G. Kang, Pseudo-Dedekind domains and divisorial ideals in R[x]T , J. Algebra 122 (1989) 323–336. [4] R. 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