IBN AL- HAITHAM J. FO R PURE & APPL. SC I . VO L.23 (2) 2010 Chained fuzzy modules S. B.Semeein Departme nt of Mathematics, College of Education I bn-Al-Haitham,Unive rsity of Baghdad Abstract Let R be a commutative ring with unity . In this p aper we introduce the notion of chained fuzzy modules as a generalization of chained modules. We invest igate several characterizations and p rop erties of this concept Introduction In this p aper we introduce the concept of chained fuzzy modules as a generalization of the concep t (chained modules) in ordinary algebra .This p aper consist s of t hree sections In section one, we recall some basic definitions and results which we needed later. In section two, we give some results about chained fuzzy modules such as it's relationship with it levels. Section three is devoted for study ing the direct sum of chained fuzzy modules. Finally , we st udy the homomorphic image and inverse of chained fuzzy modules. 1. Preliminaries The following definitions and results are needed later. 1.1 De fini tion, [1] Let M be a nonempty set and I be the closed interval [0,1] of the real line (numbers). A fuzzy set A in M (a fuzz y subset A of M ) is a function from M into I. 1.2 De fini tion, [2] Let xt:M  [0,1] be a fuzzy set in M , where xM , t[0,1] defined by: t t if x y X ( y) 0 if x y     for all yM , xt is called a fuzz y singleton or fuzzy p oint in M , if x=0 and t=1, then 1 1 if y 0 0 ( y) 0 if y 0     We shall call such fuzzy singleton the fuzz y zero singleton. 1.3 Definition , [2] Let A and B be two fuzzy sets in M , then 1. A = B i f a n d o n l y i f A ( X ) = B ( X ) , f o r a l l x  M . 2. AB if and only if A(X)  B(X), for all xM . 3. (AB)(x)=min{A(x),B(x)} for all xM . 4. (AB)(x)=max{A(x),B(x)} for all xM . 1.4 De fini tion, [3] Let A be a fuzz y set in M and t[0,1]. The set At={xM ,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]. IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (2) 2010 Where A and B are fuzzy sets. Now, we can give the definition of image and inverse image of a fuzz y set. 1.6 De fini tion, [4] Let f be a map p ing from a set M into a set N, A be a fuzzy set in M and B be a fuzz y set in N. The image defined by : 1 1sup{A(z) z f (y)} if f (y) f ( ) for all y N 0 otherwise            where f – 1 (y )={x:f(x)=y } and the inverse image of B, denoted by f – 1 (B), is the fuzz y set in M defined by: f – 1 (B)(x)=B(f(x)), for all xM . 1.7 De fini tion, [5] Let f be a map p ing 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 . The following lemma is needed in section three. 1.8 Lemma, [5] If f is a function defined on a set M , A1 and A2 are fuzzy subset of M , B1 and B2 are fuzzy subset of f(M ). Then the following are true: 1. A1=f – 1 (f(A)), whenever A1 is f-invariant. 2. f(f – 1 (B1))=B1 3. if A1A2, then f(A1)f(A2) 4. if B1B2, then f – 1 (B1)f – 1 (B2). 1.9 De fini tion, [6] 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 1. X(x+y )  min{X(x), X(y )} 2. X(x) = X(– x) 3. X(xy )  min{X(x), X(y )}. 1.10 De fini tion [7] 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 )}. 1.11 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. 1.12 De fini tion [6] Let X and A be two fuz zy modules of an R-module M . A is called a fuzzy submodule of X if AX. 1.13 Proposi tion [7] Let A be a fuzz y set of M . Then the level subset At, t(0,1] is a submodule of M if and only if A is a fuzz y submodule of X where X is a fuzzy module of M such that A(x)X(x), xM . 1.14 De fini tion [8] A fuz zy module X of an R-module M is called fuzz y simple if and only if X has no fuzzy p rop er submodules. 1.15 De fini tion [8] A fuz zy module X of an R-module M is called fuzzy cyclic module, if there exists xtX such that each y kX writt en as y k=rℓxt for some fuzz y IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (2) 2010 singleton rℓ of R where k, ℓ, t[0,1]. In this case, we shall write X=(xt) to denoted the fuzzy cyclic module generated by xt. 1.16 De fini tion [6] Let X and Y be two fuzzy 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. 1.17 Remark [9] 1- Let M and M ' be two R-modules, f:MM ' be an epimorp hism. If A is a fuzzy submodule of M , then f(A) is a fuzz y submodule of M '. 2- Let M and M ' be two R-modules, f:MM ' be a homomorp hism. If B is a fuzzy submodule of M ', then f – 1 (B) is a fuzz y submodule of M . 1.18 De fini tion [2] Sup p ose A and B be two fuzzy modules of R-module M . We define (A:B) by:- (A:B)={rt:rt is a fuzz y singleton of R such that rtBA} and (A:B)(r)=sup {t[0,1]rtBA, for all rR}. If B=(bk), (A:(bk))={rtrtbkA, rt is a fuzzy singleton of R}. 1.19 De fini tion [10] 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. 1.20 Proposi tion [10] Let X and Y are fuzzy modules of M 1 and M 2 resp ectively, then XY is a fuzz y module of M 1M 2. 1.21 Proposi tion [10] 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. Chained Fuzzy Module In this section we introduce the concept of chained fuzzy module . some basic results of this concept are considerate 2.1 De fini tion, [11] An R-module M is called chained module if for each submodules A, B of M , either A  B or B  A. We fuzzify this definition as follows: 2.2 De fini tion Let X be a fuzzy module of an R-module M then X is called a chained fuzzy module if for each fuzz y submodules of X either A  B or B  A. To p rove our next theorem, first we p rove the following lemma: 2.3 Lemma Let A and B be two fuzzy subset of R t hen A  B if and only if At  Bt, for each t[0,1]. Proof: It is easy so it is omitted. The following theorem characterizes chained fuzzy module in terms of it is level module. 2.4 The orem A fuz zy module X of an R-module M is a chained if and only if Xt is a chained module,  t(0,1]. Proof: If X is chained fuzzy module. To p rove Xt is chained module  t(0,1]. Let I, J be submodules of Xt. Define : t x (x) 0 x      , t x J B(x ) 0 x J    IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (2) 2010 A, B are fuzz y submodules of X. But At=I, Bt=J since X is chained fuzzy module, then either A  B or B  A. Hence At  Bt or Bt  At (by lemma (2.3)). Thus I  J or J  I. Conversely , if Xt is chained module, to p rove X is a chained fuzzy module, let A, B fuzzy submodules in X. Then At, Bt are submodules in Xt, for all t(0,1] since Xt is chained R- module then At  Bt or Bt  At which implies A  B or B  A (lemma (2.3)). 2.5 Examples 1. Let X(x)=1 for all xZ8 Xt=Z 8 for all t[0,1]. But Z8 is chained. Hence by theorem (2.4) X is a chained fuzzy module. 2. Every fuzzy simple module is a chained fuzzy module. 2.6 Remark If YX and X is a chained fuzzy module then Y is chained fuzzy module. Proof: Let A, B be two fuz zy submodules of Y then A, B are fuzz y submodules of X, since X is chained fuzzy module. Then AB or BA which imp lies Y is a chained fuzz y module. 2.7 De fini tion, [10] A fuz zy module X is called uniform fuzzy module if AB01 for any nontrivial fuzzy submodules A and B of X. 2.8 Proposi tion A fuz zy module X of an R-module M is uniform if and only if Xt is a uniform module,  t(0,1]. Proof: If X is uniform fuzzy module, to p rove Xt is uniform module  t(0,1]. Let I, J be submodules of Xt. Define t x (x) 0 x      , t x J B(x ) 0 x J    A, B are fuzzy submodules of X. But At=I, Bt=J since X is uniform fuzzy module, then AB01. Hence (AB)t01 which implies AtBt01 (by remark 1.5). This IJ01. Conversely , if Xt is uniform module, to p rove X is a uniform fuzzy module, let A, B fuzzy submodules in X. Then At, Bt are submodules in Xt, for all t(0,1], since Xt is uniform R- module then AtBt01 which implies (AB)t01 (by remark 1.5). Thus AB01. Now, we shall show the relationship between uniform fuzzy module and chained fuzzy module as t he following p rop osition: 2.9 Proposi tion Every chained fuzzy module is a uniform fuzzy module. Proof: Let X be a chained fuzzy module of an R-module M then AB or BA if AB then AB= A if BA t hen AB=B which implies AB=01. 2.10 Remark The converse of prop osition (2.9) is not true for the following examp le shows: 2.11 Example Let M =Z as a Z-module X:M[0,1] such that X(x)=1 xM Xt=z for all t[0,1]. But Z is uniform. Hence by p rop osition (2.9) X is a uniform fuzzy module. But X is not chained fuzzy module since  A, B fuzzy submodules of X defined by 1 x (2) (x) 1 x (2) 4      , 1 x (5) B(x ) 1 x (5) 4      and A  B and B  A. Recall that if A and B are two submodules of an R-module M , then A and B are called comp arable if AB and BA. We shall fuzzify this concept as follows: IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (2) 2010 2.12 De fini tion Let A, B be two fuz zy submodules of a fuzzy module X of an R-module M , then A and B are called comp arable if AB and BA. 2.13 Proposi tion A fuz zy module X of an R-module M is chained iff every two cy clic fuzzy submodules of X are comparable. Proof: Let A and B be fuzz y submodules of X. Sup p ose A  B, we show BA since A  B, there exists xtA and xtB A and  B. Let y kB, then B, , are cyclic fuzzy submodules of X, then either or . If implies  B (since B). Thus xtB is a contradiction. If implies that  A (since  A). T hus BA so X is chained. The converse is obvious. 2.14 Remark A chained fuzz y module is indecomposable. Proof: Sup p ose X is decomposable, then X=AB for some fuzzy submodule A and B of X. Thus AB=O1 is a contradiction (p rop osition 2.9). Now, we introduce the notion of chained fuzzy ring. First we have the following definition. 2.15 De fini tion, [11] A ring R is called chained if and only if for each fuzzy ideals I, J of R either IJ or JI. 2.16 De fini tion A fuz zy ring X of a ring R is called chained if and only if for each fuzzy ideals I, J of X either IJ or JI. 2.17 Remark A fuz zy ring X is chained if and only if Xt is chained ring  t(0,1]. Proof: It is easy so it is omitted. 2.18 De fini tion, [8] A fuzzy module X of an R-module M is called multiplication fuzzy module if for each nonemp ty fuzzy submodule A of X, there exists a fuzzy ideal I of R such that A=IX. 2.19 Proposi tion Let X be a multiplication module of an R-module M if R is a chained ring then X is a chained fuzzy module. Proof: Let A and B be fuzzy submodules of X. Then there exists fuzzy ideals I and J of R such that A=IX and B=JX, since It and Jt ideals of R and R is chained, therefore It  Jt or Jt It. Thus IJ or JI (by remark 2.3) implies that IXJX or JXIX. T hus AB or BA. 2.20 De fini tion Let X be a chained fuzzy module of an R-module M and let V(X)={(O1:Xt)XtX}. 2.21 De fini tion, [10] Let X be a non empty fuzzy module of R-module M . The fuzz y annihilator of A denoted by (F-annA) is defined by {Xt:xR,XtAO1},t[0,1], where A is a prop er fuzzy submodule of X. Not e that: (F-annA)(a)=sup {t :t[0,1],atAO1}, for all aR; that is F-annA=(O1:A). 2.22 De fini tion, [10] A fuz zy module X is called faithful if F-annX=O1. 2.23 Remark If X is chained faithful fuzzy module then 1 tO X X   (O1:Xt)=O1, XtX. IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (2) 2010 Proof: If 1 tO X X   (O1:Xt)O1 then there is rℓR, rℓO1 such that rℓxt = O1, xtX t hen rℓX=O1 a contradiction. 2.24 De fini tion, [13] A fuz zy ideal A of a ring R is called fuzz y p rime ideal, if A is non-constant and for any fuzzy ideals B and C of R such that B CA, t hen either BA or CA. Equivalently , A fuzzy ideal A of a ring R is called fuzzy p rime ideal if A is a non-constant and for all aℓ, bh fuzzy singletons of R such that aℓbhA implies that either aℓ A or bhA, ℓ, h[0,1]. 2.25 Remark If X is a chained fuzzy module of an R-module M then, 1. V(X) is a linearly ordered set of fuzzy ideals of R. 2. P= 1 tO X X   (O1:xt) is a fuzz y p rime ideal of R. Proof: (1) Let A, B  V(X) t hen A=(O1:xt) and B=(O1:yt) for some xtO1, y t=O1 and xt, y tX since X is chained fuzzy module then Xt is chained module (by theorem (2.4)) imp lies that V(Xt) is a linearly ordered set of ideals of R (see [12,remark (1.9)). Thus V(X) is a linearly ordered set of fuzzy ideals of R. (2) P= 1 tO X X   (O1:xt) is a fuzz y ideal of R. To show that P is a fuzz y p rime ideal, let aℓ, bhR such that aℓbhP, t hen there is O1 xtX such that aℓbh(O1:xt). Then aℓbh xt=O1. This imp lies that aℓ(O1: bhxt). Now if bhxt= O1 then bh(O1:xt). Thus bhP and if bhxt O1 then bhxt O1X, and hence aℓP. 3. Di rect Sum of Chained Fuzzy Modul e We turn att ention to the direct sum of chained fuzzy modules. 3.1 Remark If X and Y are two chained fuzzy modules of an R-module M 1 and M 2 resp ectively then XY is not necessary chained fuzzy module of M 1  M 2 as the following examp le shows: 3.2 Example Let X:Z6{0, 1 3 } such that 1 if a 2 X (a) 3 0 i f a , 2         Let Y:Z6{0, 1 3 } such that 1 if a 3 Y (a) 3 0 i f a , 3         It is clear that X and Y are chained fuzzy modules of Z6. Hence XY is not a chained fuzzy module of Z6Z6. Since  A,B fuz zy submodules of XY, where 1 i f (a, b) 2 0 A (a, b) 3 0 if (a, b) , 2 0              and 1 if (a, b) 0 3 B(a, b) 3 0 if (a, b) , 0 3               IHJPAS IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.23 (2) 2010 But A(2,0)= 1 3 , B(2,0)=0, that is A  B. Alaso A(0,3)=0, B(0,3)= 1 3 , that is B  A. Thus XY is not a chained fuzzy module of Z6Z6. 3.3 The orem Let X and Y be a fuzzy modules of an R-modules M 1 and M 2 resp ectively, if XY is a chained fuzzy module of M 1M 2 then X is a chained fuzz y module of M 1 and Y is a chained fuzzy module of M 2. Proof: By similar p roof of theorem (4.10) in [14]. Next, we shall indicate the behaviors of chained fuzzy modules under homomorphism. 3.4 The orem Let X and Y be a fuzzy modules of an R-modules M 1 and M 2 resp ectively. Let f:XY be a fuzzy epimorp hism. If X is a chained fuzz y module, then Y is a chained fuzz y module. Proof: Let A, B are fuzzy submodules in Y. Then f – 1 (A), f – 1 (B) are fuzzy submodules in X (remark (1.17),(2)), since X is chained fuzzy module, then either f – 1 (A) f – 1 (B) or f – 1 (B) f – 1 (A). Now, if f – 1 (A) f – 1 (B), then f(f – 1 (A)) f(f – 1 (B)) (by lemma (1.8),(2)). Similarly , if f – 1 (B) f – 1 (A), t hen BA. T herefore Y is a chained fuzzy module. 3.5 Proposi tion Let X and Y be two fuzzy modules of an R-modules M 1 and M 2 resp ectively. Let f:XY be a fuzzy homomorp hism and every submodule of Y is f-invariant. If Y is a chained fuzzy module, then X is a chained fuzzy module. Proof: Let A, B are fuzzy submodules in X. Hence f(A), f(B) are fuzzy submodules in Y (remark (1.17),(1)), since Y is a chained fuzzy module then f (A) f (B) or f (B) f (A). Now, if f (A) f (B), then f – 1 (f (A))f – 1 (f (B)) (by lemma (1.8),(4)). Hence AB (by lemma (1.8),(1)). Similarly , if f (B) f (A), t hen BA. T herefore X is a chained fuzz y module. References 1. Zahdi, L.A., (1965), Fuz zy Sets, Information and Control, 8, 338-353. 2. Zadehi, M . M ., (1992), On L-Fuzzy Residual Quot ient M odules and p -Primary Submodules, Fuz zy Sets and Sy st ems, 51:, 331-344. 3. M ashinchi, M . and Zahedi, M . M ., “On L-Fuz zy Primary Submodules”, Fuz zy Sets and Sy st ems,.49:, pp .231-236, (1996). 4. 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M ukhrjee, T.K., (1989), Prime Fuz zy Ideals in Rings, Fuz zy Sets and Sy st ems, 32:, 337-341. 14. Hadi, I.M .A., (2002), Some Ty p es of Fuz zy Rings, M athematics and Phy sics J. I(17)(1), 1-17. IHJPAS 2010) 2( 23مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة المجلد المودیوالت الضبابیة المسلسلة شروق بهجت جامعة بغداد، ابن الهیثم -كلیة التربیة، قسم الریاضیات الخالصة حلقة أبدالیة ذا عنصر محاید Rلتكن لقـد أعطینـا العدیـد . في هذا البحث قـدمنا مفهـوم المودیـوالت الضـبابیة المسلسـلة تعمیمـا لمفهـوم المودیـوالت المسلسـلة .من التمیزات والخواص األساسیة لهذا المفهوم IHJPAS