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 والحلقـــات الحســـابیة المقاســـات التوزیعیـــة الضــبابیةودرســنا قـــدمنافـــي هــذا البحـــث . محایـــدبحلقـــة ابدالیــة Rلــتكن .المفاهیموأعطینا بعض الخواص االساسیة حول هذه .للمقاسات التوزیعیة والحلقات الحسابیة) ناعتیادی(ن ییمتعمالضبابیة IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 Fuzzy Distributive Mo dules I.M.A.Hadi, Sh. B.Seme ein Departme nt of Mathematics, College of Education ,Ibn-Al-Haitham, Unive rsity of Baghdad Recei ved in December,13,2009 Accepted in March, 9,2010 Abstract Let R be a commutative ring with unity . In this p aper we introduce and st udy fuzzy distributive modules and fuzzy arithmetical rings as generalizations of (ordinary ) distributive modules and arithmetical ring. We give some basic p rop erties about t hese concep ts. Introduction In this p aper 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 st udy the direct sum of fuzzy distributive modules. In section three, we st udy the homomorp hic image and inverse image of fuzzy distributive modules. In section four, we introduce and st udy fuzzy arithmetical rings as a generalization of the concep t (arithmetical rings) in ordinary algebra. 1. Preliminaries In this section, some basic definitions and results are collected. 1.1 De fini tion [1] Let S be a non-empty set and I be the closed interval [0,1] of the real line (real numbers). A fuz zy set A in S (a fuzz y subset of S) is a function from S into I. 1.2 De fini tion [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 De fini tion [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 De fini tion [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 p rop erties 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 De fini tion [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 De fini tion [6] A fuz zy 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 De fini tion [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 De fini tion [4] Let X and A be two fuz zy modules of an R-module M . A is called a fuzz y submodule of X if AX. 1.10 proposi tion [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 De fini tion [8] Let X:R[0,1] be a fuzz y 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 De fini tion [9] Let A, B be two fuz zy ideals of a fuzzy ring X. T hen 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 Proposi tion Let A and B be two fuzzy submodules of a fuzzy module X .Then (AB)t = AtBt ,  t(0,1]. Proof : by similar p roof in [10,theorem 2.4 ] . 1.14 Proposi tion Let A and B be two fuzzy submodules of fuz zy 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 sup rimum prop erty , so t here 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 t hat 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 rep resentation of any element of M is unique. 1.15 De fini tion [2] Sup p ose that A and B be two fuz zy modules of R-modules M . We define (A:B) by:- (A:B)={r1:r1 is a fuzz y 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 fuzz y singleton of R} 1.16 De fini tion [11] Let X and Y be two fuzzy modules of M 1, M 2 resp ectively. 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 Prop osition [11] Let X and Y are fuzzy modules of M 1 and M 2 resp ectively, then XY is a fuzzy module of M 1M 2. 1.18 Proposi tion [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 Di stributi ve Modul e In this section we fuzzy ify the concep t of distributive modules into fuzzy distributive modules. Then we study some of their basic prop erties. 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 De fini tion 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 exp lains the relationship between fuzzy distributive modules and its level. 2.2 The orem 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 fuz zy distributive module. To p rove 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 p rop osition 1.13) At(Bt+C t) = (AtB t) +(AtC t) (remark 1.5 and p rop osition 1.13) This I(J+K) = (IJ)+(IK) Conversely , if Xt is a distributive module, for all t(0,1]. To p rove X is a fuzzy distributive module. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. 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 dist ributive 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 p rop osition 1.13) [A(B+C)]t = [(AB)+(AC)]t (remark 1.5 and p rop osition 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 imp lies Xt is not a distributive module. thus X is not a fuzz y distributive module. 2.4 De fini tion [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 concep t as follows. 2.5 De fini tion Let X be a fuzzy module of an R-module M then X is called a fuzzy chained module if for each fuzz y 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 Proposi tion 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 prop osition (2.6) is not true in general as the following examp le 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 The orem 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]. Pro of : 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 fuz zy 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 imp lies Y is a fuzz y distributive.  Now, we st udy the direct sum of fuzzy distributive modules. But first we st ate and prove the following lemma. 2.11 Lemma If M 1, M 2 are distributive R-modules such that annM 1 + annM 2=R, then M 1M 2=M is a dist ributive R-module. Proof: Let A, B, C be submodules of M . Since annM 1 + annM 2=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 p rove A(B+C) = (AB)+(AC) A(B+C) = (A 1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 M 2 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 Proposi tion Let X and Y be fuzzy distributive modules of R-modules M 1, M 2 resp ectively, then XY is a fuzz y distributive module of M 1M 2, p rovided annM 1 + annM 2 = R. Proof: By theorem (2.2), Xt and Yt are distributive submodules of M 1 and M 2 resp ectively, 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 Fuz zy Dist ributive M odules In this section, we shall indicate the behaviour of fuzzy distributive modules under homomorp hisms. To do this we need some definitions and prop ositions. 3.1 De fini tion (5) Let f be a map p ing from a set M into a set N, let A be a fuzz y set in M and B be a fuzzy set in N. The image of A denoted by f(A) is t he 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 De fini tion (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 De fini tion (4) Let X and Y be two fuz zy modules of R-modules M 1 and M 2 resp ectively. f:XY is called a fuzzy homomorp hism if f: M 1  M 2 is R-homomorp hism and Y(f(x)) = X(x), for each xM 1. 3.4 Proposi tion Let X and Y be two fuzzy modules of R-modules M 1 and M 2 resp ectively. f:XY be a fuzz y homomorp hism if A and B are two fuz zy submodules of X and Y resp ectively, 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 Proposi tion Let X and Y be two fuzzy modules of R-modules M 1 and M 2 resp ectively. Let f:XY be a fuzzy epimorp hism, and every fuzzy submodule of X is f-invariant. If X is a fuzzy distributive module, then Y is a fuzz y 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 p rop osition 3.4, (2). Since X is a fuzz y 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)), p rop osition 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)), p rop osition 3.4,(3),(4)) A(B+ C) = (AB) + (AC), p rop osition 3.4,(6)).  3.6 Proposi tion Let X and Y be two fuzzy modules over R-modules M 1 and M 2 resp ectively. Let f:XY be a fuzzy homomorp hism, and every fuzzy submodule of Y is f-invariant. If Y is a fuzzy distributive module, then X is a fuzz y 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 p rop osition 3.4,(1). Since Y is a fuzz y 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)), p rop osition 3.4,(3),(5)) f(A(B+ C)) = f((AB) + (AC)), p rop osition 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), p rop osition 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 De fini tion[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 De fini tion 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 fuz zy 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=Z 4 ,  xϵ Z4 .But Z4 is arithmetical ring . Hence By Not e ( 4.3 ) , X is fuzzy arithmetical ring Now, we can give the following theorem. 4.5 The orem A fuz zy 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), prop osition (1.13)) (A + (B  C))t=((A+B)t  (A+C))t (remark (1.5), prop osition (1.13)) Thus A + (B  C) = (A+B)  (A+C). Conversely , to p rove X is a fuzz y arithmetical ring. We shall p rove 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 hy p othesis, 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), p rop osition (1.13)). But At=I, Bt=J, Ct=K, hence I(J+K)=(IJ)+(IK), which imp lies that Xt is an arithmetical ring, by ((16), Exc.18). Thus X is an arithmetical ring, by note (4.3).  4.6 The orem 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 fuzz y ideals A, B of X. Proof: (1)  (2): to p rove A(BC) = AB  AC for all fuzz y 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, (p rop osition (1.12),(2), remark (1.5)) (A(BC))t = (ABAC)t, for all t (0,1] (p rop osition 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 fuz zy 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 imp lies 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),p rop osition (1.12)), so that At(BtCt) = AtBt  AtCt, (remark (1.5),prop osition (1.12)). I (JK)=(IJ)+(IK). T hen 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 fuzz y 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 imp lies that ((A+B)(AB))t=(AB)t for all t (0,1]. Hence (A+B)t(AB)t=AtBt, (p rop osition (1.12),(2)), so that (At+Bt) (AtBt) =AtBt, (remark (1.5), prop osition (1.13)). (J+K) (IJ)=IJ. T hen by ([14], theorem (6.6)) Xt is arithmetical ring for all t (0,1]. Thus X is a fuzz y arithmetical ring (by note 4.3).  4.7 The orem 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 fuzz y ideals A, B of X. 4. If A, C are fuzz y ideals of X and if CA, t hen there exists fuz zy 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, t hen 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 fuzz y arithmetical ring. Re ferences 1. Zahdi, L.A. (1965), Fuz zy Sets,Information and Control, 8, 338-353. 2. Zadehi, M . M .(1992), On L-Fuz zy Residual Quotient M odules and p -Primary Submodules, Fuz zy Sets and Sy st ems, 51, 331-344. 3. Zadehi, M . M . (1991), A Characterization of L-Fuzzy Prime Ideals, Fuz zy Sets and Sy st ems, 44, 147-160. 4. M artinez, L. (1996), Fuz zy M odules Over Fuzzy Rings in Connection with Fuzzy Ideal of Fuz zy Ring, J. Fuzzy M ath., 4, 843-857. 5. Nanda, S. (1989), Fuz zy M odules Over Fuzzy Rings, Bull. Col.M ath. Soc., 81, 197-200. 6. Bhambert, S.K.; Kumar and Kumar P. (1995), Fuz zy Prime Submodule and Radical of Fuz zy Submodules, Bull. Soc., 87, 163-168. 7. M ukherjee, T. K.; Sen, M . K. and Roy , D. (1996), n Fuzzy Submodules and their Radicals, J. Fuzzy M ath., 4, 549-558. 8. Liu, W.J. (1982), Fuz zy Invariant Subgroup s and Fuz zy Ideals, Fuz zy Sets and Sy st ems, 8, 133-139. 9. M artine, L. (1995), Fuz zy Subgroup s of Fuz zy Group s and Fuzzy Ideals of Fuz zy Ring, The Journal of Fuz zy M ath., 3:(4), 883-849. 10. Hadi, M .A. (2001), On Fuz zy Ideals of Fuz zy Rings, 16:(4), 17-33. 11. Rabi, H. J. (2001), Prime Fuz zy Submodules and Prime Fuzzy M odules, M .Sc. Thesis, University of Baghdad. 12. M ohmad, A.A. (1997), Chained M odules, M .Sc. Thesis University of Baghdad. 13. Shores, T.S. and Lewis, W.J. (1974), Serial M odules and Endomorphism Rings, Duke M ath. J., 41, 889-909. 14. Kumar, R. (1991), Fuz zy Semi-Primary Ideals of Rings, Fuz zy Sets and Sy st ems, 42, 263-272. 15. Zhao, Jiandi, Shik. Yue M . (1993), Fuz zy M odules Over Fuzzy Rings, T he J. of Fuzzy M ath., 3, 531-540. 16. Larson, M .D. and M ccarthy , P.J. (1971), M ultip licutive Theory of Ideals, Academic Press, London, New Yourk.