2011) 1( 24مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد المقاسات التوزیعیة الضبابیة شروق بهجت ، أنعام محمد علي هادي جامعة بغداد،ابن الهیثم -كلیة التربیة ،قسم الریاضیات 2009 كانون االول 13استلم البحث في 2010 اذار 9قبل البحث في الصةالخ ــة ابدالیــة Rلــتكن ـبابیةودرســنا قـــدمنافـــي هــذا البحـــث . محایـــدبحلقـ ـات التوزیعیـــة الضـ ـابیة المقاســ ـات الحســ والحلقــ .المفاهیموأعطینا بعض الخواص االساسیة حول هذه .للمقاسات التوزیعیة والحلقات الحسابیة) ناعتیادی(ن ییمتعمالضبابیة IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (1) 2011 Fuzzy Distributive Modules I.M.A.Hadi, Sh. B.Semeein Department of Mathematics, College of Education ,Ibn-Al-Haitham, University of Baghdad Received in December,13,2009 Accepted in March, 9,2010 Abstract Let R be a commutative ring with unity . In this paper we introduce and study fuzzy distributive modules and fuzzy arithmetical rings as generalizations of (ordinary) distributive modules and arithmetical ring. We give some basic properties about these concepts. Introduction In this paper we introduce and study fuzzy distributive modules as a generalization of the concept (distributive modules) in ordinary algebra. In section one, we recall some basic definitions and results which we will be needed later. In section two, we give some basic results about fuzzy distributive modules. Also we study the direct sum of fuzzy distributive modules. In section three, we study the homomorphic image and inverse image of fuzzy distributive modules. In section four, we introduce and study fuzzy arithmetical rings as a generalization of the concept (arithmetical rings) in ordinary algebra. 1. Preliminaries In this section, some basic definitions and results are collected. 1.1 Definition [1] Let S be a non-empty set and I be the closed interval [0,1] of the real line (real numbers). A fuzzy set A in S (a fuzzy subset of S) is a function from S into I. 1.2 Definition [2] Let xt:S  [0,1] be a fuzzy set in S, where xS, t[0,1] defined by: Xt(y)=t if x=y, and xt(y) = 0 if xy yS. Xt is called a fuzzy singleton or fuzzy point in S. 1.3 Definition [3] Let A and B be two fuzzy sets in S, then 1. A=B if and only if A(X)=B(X), for all xS. 2. AB if and only if A(X)  B(X), for all xS. 3. (AB)(x)=min{A(x),B(x)} for all xS. 1.4 Definition [4] Let A be a fuzzy set in S, for all t[0,1], the set At={xS,A(x)t} is called level subset of A. 1.5 Remark [1] The following properties of level subsets hold for each t(0,1] 1. (AB)t=At  Bt 2. A=B if and only if At=Bt, for all t(0,1]. 1.6 Definition [5] Let (R,+,) be a ring and let X be a fuzzy set in R. Then X is called a fuzzy ring in ring (R,+,) if and only if, for each x, y  R IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 1. X(x+y)  min{X(x), X(y)} 2. X(x) = X(– x) 3. X(xy)  min{X(x), X(y)}. 1.7 Definition [6] A fuzzy subset X of a ring R is called a fuzzy ideal of R, if for each x, y  R 1. X(x–y)  min{X(x), X(y)} 2. X(xy)  max{X(x), X(y)}. 1.8 Definition [2] Let M be an R-module. A fuzzy set X of M is called a fuzzy module of M if 1. X(x–y)  min{X(x), X(y)}, for all x, y M. 2. X(rx) X(x), for all xM and rR. 3. X(0)=1. 1.9 Definition [4] Let X and A be two fuzzy modules of an R-module M. A is called a fuzzy submodule of X if AX. 1.10 proposition [7] Let A be a fuzzy set of an R-module M. Then the level subset At, t[0,1] is a submodule of M if and only if A is a fuzzy submodule of X where X is a fuzzy module of an R-module M. 1.11 Definition [8] Let X:R[0,1] be a fuzzy ring, let A:R[0,1]. A is called a fuzzy ideal of X if A satisfies the following: 1. A≠ 2. A(x–y)  min{A(x), A(y)}, for all x, y R. 3. A(xy)  min{X(x), A(y)}, for all x, y R. 4. A(x)  X(x),  x  R. 1.12 Definition [9] Let A, B be two fuzzy ideals of a fuzzy ring X. Then 1. The sum A+B of A and B is defined as: (A+B)(x) = a b x sup   {min{A(a),B(b)}, xR. 2. The product AB of A and B is defined as (AB)(x) = n i i i 1 x a b sup   {inf{min{A(ai),B(bi)}}. 1.13 Proposition Let A and B be two fuzzy submodules of a fuzzy module X .Then (AB)t = AtBt ,  t(0,1]. Proof : by similar proof in [10,theorem 2.4 ] . 1.14 Proposition Let A and B be two fuzzy submodules of fuzzy module. Then (A+B)t = At+Bt,  t(0,1]. Proof:- Let x(A+B)t. Then (A+B)(x)=sup{min{A(a),B(b)}, x=a+b}t But A+B has a suprimum property , so there exist a, b M such that sup{min{A(a),B(b)}, x=a+b}=min{A(a),B(b)}t consequently, A(a)t, B(b)t. Thus, aAt and bBt, it follows that x=a+bAt+Bt which means (A+B)t  At+Bt Now, let xAt+Bt, then ! aAt and ! bBt such that x= a+b. Thus (A+B)(x)=sup{min{A(a),B(b)},x=a+b} IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 = min{A(a),B(b)}t Since the representation of any element of M is unique. 1.15 Definition [2] Suppose that A and B be two fuzzy modules of R-modules M. We define (A:B) by:- (A:B)={r1:r1 is a fuzzy singleton of R such that r1BA} and (A:B)(r)=sup{ t[0,1]  rtBA, for all rR} If B=(bk), then: (A:(bk))={rtrtbk A, rt is a fuzzy singleton of R} 1.16 Definition [11] Let X and Y be two fuzzy modules of M 1, M 2 respectively. Define XY: M 1M 2[0,1] by (XY)(a,b)=min{X(a),Y(b)} for all (a,b)  M 1M 2 XY is called a fuzzy external direct sum of X and Y. 1.17 Proposition [11] Let X and Y are fuzzy modules of M 1 and M 2 respectively, then XY is a fuzzy module of M1M 2. 1.18 Proposition [11] Let A and B be two fuzzy submodules of a fuzzy module X, such that X=AB, then Xs = As Bs for all s(0,1]. 2. Fuzzy Distributive Module In this section we fuzzyify the concept of distributive modules into fuzzy distributive modules. Then we study some of their basic properties. Recall that an R-module M is said to be distributive if for any R-submodules A, B and C of M, A(B+C) = (AB)+(AC) [12]. 2.1 Definition Let M be an R-module, let X be a fuzzy module over M. X is called distributive if for any fuzzy submodules A, B and C of X, A(B+C) = (AB)+(AC) The following result explains the relationship between fuzzy distributive modules and its level. 2.2 Theorem A fuzzy module X of an R-module M is a fuzzy distributive if and only if Xt is a distributive module, t(0,1]. Proof: If X is fuzzy distributive module. To prove Xt is distributive module.  t(0,1], let I, J, K be submodules of Xt. Define t x (x) 0 x       , t x J B(x) 0 x J     , t x K C(x) 0 x K     It is clear that A, B, C are fuzzy submodules of X and At=I, Bt=J, Ct=K. Since X is fuzzy distributive, A(B+C) = (AB)+(AC). Hence [A(B+C)]t = [(AB)+(AC)]t,  t(0,1]. At(B+C)t = (AB)t+(AC)t (remark 1.5 and proposition 1.13) At(Bt+C t) = (AtB t) +(AtC t) (remark 1.5 and proposition 1.13) This I(J+K) = (IJ)+(IK) Conversely, if Xt is a distributive module, for all t(0,1]. To prove X is a fuzzy distributive module. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (1) 2011 Let A, B and C fuzzy submodules in X. Then At, Bt, Ct are submodules in Xt, for all t(0,1]. Since Xt is a distributive R-module then At(Bt+C t) = (AtB t) +(AtC t) At(B+C)t = (AB)t+(AC)t (remark 1.5 and proposition 1.13) [A(B+C)]t = [(AB)+(AC)]t (remark 1.5 and proposition 1.13) Then A(B+C) = (AB)+(AC). (remark 1.5 ,(2) )  2.3 Example Let M=RR where R is any ring, M is an R-module, let X:M [0,1] defined by X(x)=1, let 1 (x, y) R(1,1) (x,y) 0 otherwise      , 1 (x,y) R(0,1) (x, y) 0 otherwise      , 1 (x,y) R(1,0) C(x, y) 0 otherwise     At=R(1,1), Bt=R(0,1), Ct=R(1,0),  t(0,1]. At(Bt+C t) =R(1,1), (AtB t)+(AtCt)=(R(1,1)R(0,1))+(R(1,1)R(1,0))=(0)+(0)=(0) Thus At(B+C)t  (AB)t+(AC)t, which implies Xt is not a distributive module. thus X is not a fuzzy distributive module. 2.4 Definition [13] An R-module M is called chained if for each submodules A, B of M, either A  B or B  A. We fuzzified this concept as follows. 2.5 Definition Let X be a fuzzy module of an R-module M then X is called a fuzzy chained module if for each fuzzy submodules A, B of X, either A  B or B  A. Now, we shall give a relationship between fuzzy distributive module and fuzzy chained module. 2.6 Proposition Let X be a fuzzy chained module of an R-module M. Then X is a fuzzy distributive module. Proof: Let A, B, C fuzzy submodules of X, we can assume that ABC. Hence A(B+C)=AB=A. But (AB)+(AC)=A+A=A. Thus A(B+C) = (AB)+(AC).  2.7 Remark The converse of proposition (2.6) is not true in general as the following example shows. 2.8 Example Let X(x)=1 for all xZ, Xt=Z, t[0,1]. But Z is distributive. Hence by theorem (2.2), X is a fuzzy distributive. However X is not chained since there exists fuzzy submodules A, B such that 1 x 2Z (x) 0 x 2Z     , 1 x 3Z (x) 0 x 3Z      and A  B and B  A. Now, we can give the following. 2.9 Theorem Let X be a fuzzy distributive module of an R-module M, then for all at, bk X, <1j>=(at:bk)+(bk:at) for all j(0,1]. Proof: Let at, bk X, then aXt, bXk. Assume kt. Hence Xk  Xt and so aXk. Thus a, b  Xk. But Xk is distributive R-module so (a:b)+(b:a)=R (by [12,theorem (1.3),p.54). It follows that 1=r1+r2 where r1(a:b), r2(b:a) for some r1, r2. Hence 1j= (r1)j + (r2)j for all j(0,1]. But IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 (r1)jbk = (r1b)s , where s = min{j,k} = (r'a)s , since r1(a:b)  < as>  < ak > Hence (r1)j(at:bk),  j(0,1] (r2)jat = (r2a)f , where f = min{j,t} = (r''b)f , since r2(b:a)  < bt>  < bk > Hence (r2)j(bk:at),  j(0,1] So (r1)j+(r2)j(at:bk) + (bk:at) and hence (r1+r2)j(at:bk) + (bk:at). Thus 1j  (at:bk) + (bk:at).  2.10 Remark If YX and X is fuzzy distributive module then Y is fuzzy distributive module. Proof: Let A, B, C are fuzzy submodules of Y, then A, B, C are fuzzy submodules of X (since YX). But X is fuzzy distributive module, so A(B+C) = (AB)+(AC) which implies Y is a fuzzy distributive.  Now, we study the direct sum of fuzzy distributive modules. But first we state and prove the following lemma. 2.11 Lemma If M 1, M 2 are distributive R-modules such that annM1 + annM2=R, then M1M 2=M is a distributive R-module. Proof: Let A, B, C be submodules of M. Since annM 1 + annM2=R, A=A1B1, B= A2B2, C= A3B3 for some submodules A1, A2, A3 of M 1 and some submodules B1, B2, B3 of M 2. To prove A(B+C) = (AB)+(AC) A(B+C) = (A1B1)[(A2B2)+(A3B3)] = (A1B1)[(A2+A3)+( B2+B3)] = [A1(A2+A3)][B1(B2+B3)] = [(A1A2)+(A1A3)][(B1B2)+(B1B3)] (M 1 and M2 are distributive modules) = [(A1A2)(B1B2)] + [(A1A3)(B1B3)] = [(A1B1)(A2 B2)]+[(A1B1)(A3 B3)] = (AB)+(AC).  2.12 Proposition Let X and Y be fuzzy distributive modules of R-modules M 1, M 2 respectively, then XY is a fuzzy distributive module of M 1M 2, provided annM1 + annM2 = R. Proof: By theorem (2.2), Xt and Yt are distributive submodules of M 1 and M2 respectively, for all t (0,1]. Hence by lemma (2.11) (XtYt) is a distributive submodule of M 1M 2. But (XY)t = (XtYt) by ((11), lemma (2.2.4)). Thus XY is a fuzzy distributive module by theorem (2.2).  3. The Image and Inverse Image of Fuzzy Distributive Modules In this section, we shall indicate the behaviour of fuzzy distributive modules under homomorphisms. To do this we need some definitions and propositions. 3.1 Definition (5) Let f be a mapping from a set M into a set N, let A be a fuzzy set in M and B be a fuzzy set in N. The image of A denoted by f(A) is the set in N defined by 1 1sup{A(z) z f (y)} if f (y) , y A f(A)(y) 0 otherwise          And the inverse image of f denoted by f – 1 (B), where f –1(B)(x)=B(f(x)), for all xM. Recall the following. IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 3.2 Definition (14) Let f be a function from a set M into a set M'. A fuzzy subset A of M is called f-invariant if A(x)=A(y) whenever f(x)=f(y), where x, y M. 3.3 Definition (4) Let X and Y be two fuzzy modules of R-modules M 1 and M2 respectively. f:XY is called a fuzzy homomorphism if f: M 1  M 2 is R-homomorphism and Y(f(x)) = X(x), for each xM 1. 3.4 Proposition Let X and Y be two fuzzy modules of R-modules M 1 and M 2 respectively. f:XY be a fuzzy homomorphism if A and B are two fuzzy submodules of X and Y respectively, then 1. f(A) is a fuzzy submodule of Y, (14). 2. f – 1(A) is a fuzzy submodule of Y, (14). 3. f(AB)=f(A)f(B), whenever A, B are f-invariant, (15). 4. f – 1 (AB)=f – 1 (A)f – 1 (B), where f is monomorphism, (15). 5. f(A+B)=f(A)+f(B), (15). 6. f(f – 1(A))=A, (15). 7. f– 1(f(A))=A whenever A is f-invariant, (15). First we have the following result. 3.5 Proposition Let X and Y be two fuzzy modules of R-modules M 1 and M2 respectively. Let f:XY be a fuzzy epimorphism, and every fuzzy submodule of X is f-invariant. If X is a fuzzy distributive module, then Y is a fuzzy distributive module. Proof: Let A, B, C be fuzzy submodules in Y. f – 1(A), f– 1(B), f– 1(C) are fuzzy submodules in X by proposition 3.4, (2). Since X is a fuzzy distributive, then f– 1(A)(f– 1(B)+ f– 1(C)) = (f– 1(A)f– 1(B)) + (f– 1(A)f– 1(C)) f – 1 (A)(f – 1 (B)+ f – 1 (C)) = (f – 1 (AB)) + (f – 1 (AC)), proposition 3.4,(4)) f[f – 1 (A)(f – 1 (B)+ f – 1 (C))]=f[(f – 1 (AB)) + (f – 1 (AC))] f(f – 1 (A))f(f – 1 (B)+ f – 1 (C)) = f(f – 1 (AB)) + f(f – 1 (AC)), proposition 3.4,(3),(4)) A(B+ C) = (AB) + (AC), proposition 3.4,(6)).  3.6 Proposition Let X and Y be two fuzzy modules over R-modules M 1 and M 2 respectively. Let f:XY be a fuzzy homomorphism, and every fuzzy submodule of Y is f-invariant. If Y is a fuzzy distributive module, then X is a fuzzy distributive module. Proof: Let A, B, C are fuzzy submodules in X. Hence f(A), f(B), f(C) are fuzzy submodules in Y, by proposition 3.4,(1). Since Y is a fuzzy distributive module, then f(A)(f(B)+ f(C)) = (f(A)f(B)) + (f(A)f(C)) f(A)(f(B)+ f(C)) = f(AB) + f(AC)), proposition 3.4,(3),(5)) f(A(B+ C)) = f((AB) + (AC)), proposition 3.4,(3),(5)). f – 1(f(A(B+ C))) = f– 1(f((AB) + (AC))). Then A(B+ C) = (AB) + (AC), proposition 3.4,(7)).  4. Fuzzy Arithmetical Rings In this section, we introduce the notion of arithmetical fuzzy ring. First we have the following definition. 4.1 Definition[12] A ring R is said to be an arithmetical ring if R, considered as R-module over it self, is distributive that is R is arithmetical if I(J+K)=(IJ)+(IK) for all ideals I, J, K of R. We fuzzify this definition as follows: 4.2 Definition A fuzzy ring X of a ring R is called arithmetical if and only if A(B+ C) = (AB) + (AC) for all A, B, C fuzzy ideals of X. IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 4.3 Note A fuzzy ring X is arithmetical if and only Xt is arithmetical ring t(0,1]. 4.4 example Let X(x)=1 for all xϵ Z4 , Xt=Z4 ,  xϵ Z4 .But Z4 is arithmetical ring . Hence By Note ( 4.3 ) , X is fuzzy arithmetical ring Now, we can give the following theorem. 4.5 Theorem A fuzzy ring X of a ring R is arithmetical if and only if A + (B  C) = (A+B)  (A+C) Proof: Let X be a fuzzy arithmetical ring, let A, B, C be fuzzy ideals of X. Hence At, Bt, Ct are ideals of Xt. Since X is fuzzy arithmetical ring then Xt is arithmetical ring (by note (4.3)). Hence, At + (Bt  C t) = (At + Bt) (At + Ct) ((16),Exc.18) At + (B  C)t = (A+B)t  (A+C) t, (remark (1.5), proposition (1.13)) (A + (B  C))t=((A+B)t  (A+C))t (remark (1.5), proposition (1.13)) Thus A + (B  C) = (A+B)  (A+C). Conversely, to prove X is a fuzzy arithmetical ring. We shall prove Xt is an arithmetical ring for all t(0,1]. Let I, J, K be ideals in Xt. It follows that there exist A, B, C fuzzy ideals of X, where t x I (x) 0 x I      , t x J (x) 0 x J      , t x K c(x) 0 x K     But by hypothesis, A + (B  C) = (A+B)  (A+C). Hence [A + (B  C)]t = [(A+B)  (A+C)]t for all t(0,1]. It follows that; At + (Bt  Ct) = (At+Bt)  (At+Ct), (remark (1.5), proposition (1.13)). But At=I, Bt=J, Ct=K, hence I(J+K)=(IJ)+(IK), which implies that Xt is an arithmetical ring, by ((16), Exc.18). Thus X is an arithmetical ring, by note (4.3).  4.6 Theorem Let R be an integral domain, let X be a fuzzy ring such that X(a)=1 aR. Then the following are equivalent 1. X is arithmetical 2. A(BC)=AB  AC for all fuzzy ideals A, B, C of X. 3. (A+B)(AB)=AB for all fuzzy ideals A, B of X. Proof: (1)  (2): to prove A(BC) = AB  AC for all fuzzy ideals A, B, C of X. Since X is a fuzzy arithmetical then Xt is an arithmetical ring for all t (0,1] and since At, Bt, Ct are ideals of Xt, t (0,1] we get At(BtCt) = AtBt  AtCt, ([14],theorem (6.6)). Hence At(BC)t = (AB)t  (AC)t, (proposition (1.12),(2), remark (1.5)) (A(BC))t = (ABAC)t, for all t (0,1] (proposition 1.12,(2)) Thus A(BC)=AB  AC. (2)  (3): If A(BC)=AB  AC for all fuzzy ideals A, B,C of X, let t (0,1], let I, J, K be ideals of Xt. Then there exists fuzzy ideals A, B, C of X such that At=I, Bt=J, Ct=K, where t x I (x) 0 x I      , t x J (x) 0 x J      , t x K c(x) 0 x K     By (2), A(BC)=(AB)(AC), which implies that (A(BC))t = (ABAC)t, for all t (0,1]. Hence At(BC)t = (AB)t  (AC)t, (remark (1.5),proposition (1.12)), so that At(BtCt) = AtBt  AtCt, (remark (1.5),proposition (1.12)). I (JK)=(IJ)+(IK). Then by ((16),Exc. 18) (I+J)(IJ)=IJ. Thus (At+Bt) (AtBt) = AtBt, (A+B)t (AB)t = (AB)t which implies that (A+B)(AB)=AB. (3)  (1):If (A+B)(AB)=AB for all fuzzy ideals A, B of X. Let t (0,1], Let I, J, K be IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1) 2011 ideals of Xt. Then there exists fuzzy ideals A, B, C of X such that At=I, Bt=J, Ct=K, where t x I (x) 0 x I      , t x J (x) 0 x J      , t x K c(x) 0 x K     By (3), (A+B)(AB)=AB, which implies that ((A+B)(AB))t=(AB)t for all t (0,1]. Hence (A+B)t(AB)t=AtBt, (proposition (1.12),(2)), so that (At+Bt) (AtBt) =AtBt, (remark (1.5), proposition (1.13)). (J+K) (IJ)=IJ. Then by ([14], theorem (6.6)) Xt is arithmetical ring for all t (0,1]. Thus X is a fuzzy arithmetical ring (by note 4.3).  4.7 Theorem Let R be a noetherian integral domain, let X be a fuzzy ring such that X(a)=1 aR. Then the following are equivalent 1. X is arithmetical 2. A(BC)=AB  AC for all fuzzy ideals A, B, C of X. 3. (A+B)(AB)=AB for all fuzzy ideals A, B of X. 4. If A, C are fuzzy ideals of X and if CA, then there exists fuzzy ideal B such that A=BC. Proof: (1)  (2) and (2)  (3) follows directly by theorem (4.5). (3)  (4), let A, C be fuzzy ideals of X such that CA, implies At  Ct, then At = BtCt ([14], theorem (6.26)), At=(BC)t then A = BC. (4)  (1), let A, C are fuzzy ideals of X, if CA, then there exists fuzzy ideal B of X such that A=BC, then At=(BC)t so w get At=BtCt , Xt is arithmetical ring ([14], theorem (6.26)). Thus X is a fuzzy arithmetical ring. References 1. Zahdi, L.A. (1965), Fuzzy Sets,Information and Control, 8, 338-353. 2. Zadehi, M. M.(1992), On L-Fuzzy Residual Quotient Modules and p-Primary Submodules, Fuzzy Sets and Systems, 51, 331-344. 3. Zadehi, M. M. (1991), A Characterization of L-Fuzzy Prime Ideals, Fuzzy Sets and Systems, 44, 147-160. 4. Martinez, L. (1996), Fuzzy Modules Over Fuzzy Rings in Connection with Fuzzy Ideal of Fuzzy Ring, J. Fuzzy Math., 4, 843-857. 5. Nanda, S. (1989), Fuzzy Modules Over Fuzzy Rings, Bull. Col.Math. Soc., 81, 197-200. 6. Bhambert, S.K.; Kumar and Kumar P. (1995), Fuzzy Prime Submodule and Radical of Fuzzy Submodules, Bull. Soc., 87, 163-168. 7. Mukherjee, T. K.; Sen, M. K. and Roy, D. (1996), n Fuzzy Submodules and their Radicals, J. Fuzzy Math., 4, 549-558. 8. Liu, W.J. (1982), Fuzzy Invariant Subgroups and Fuzzy Ideals, Fuzzy Sets and Systems, 8, 133-139. 9. Martine, L. (1995), Fuzzy Subgroups of Fuzzy Groups and Fuzzy Ideals of Fuzzy Ring, The Journal of Fuzzy Math., 3:(4), 883-849. 10. Hadi, M.A. (2001), On Fuzzy Ideals of Fuzzy Rings, 16:(4), 17-33. 11. Rabi, H. J. (2001), Prime Fuzzy Submodules and Prime Fuzzy Modules, M .Sc. Thesis, University of Baghdad. 12. Mohmad, A.A. (1997), Chained Modules, M .Sc. Thesis University of Baghdad. 13. Shores, T.S. and Lewis, W.J. (1974), Serial Modules and Endomorphism Rings, Duke Math. J., 41, 889-909. 14. Kumar, R. (1991), Fuzzy Semi-Primary Ideals of Rings, Fuzzy Sets and Systems, 42, 263-272. 15. Zhao, Jiandi, Shik. Yue M. (1993), Fuzzy Modules Over Fuzzy Rings, The J. of Fuzzy Math., 3, 531-540. 16. Larson, M .D. and Mccarthy , P.J. (1971), Multiplicutive Theory of Ideals, Academic Press, London, New Yourk.