IBN AL- HAITHAM J. FO R PURE & APPL. SC I . VO L.23 (2) 2010 Chained fuzzy modules S. B.Semeein Department of Mathematics, College of Education Ibn-Al-Haitham,University of Baghdad Abstract Let R be a commutative ring with unity . In this paper we introduce the notion of chained fuzzy modules as a generalization of chained modules. We investigate several characterizations and properties of this concept Introduction In this paper we introduce the concept of chained fuzzy modules as a generalization of the concept (chained modules) in ordinary algebra .This paper consists of three 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 studying the direct sum of chained fuzzy modules. Finally , we study the homomorphic image and inverse of chained fuzzy modules. 1. Preliminaries The following definitions and results are needed later. 1.1 Definition, [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 fuzzy subset A of M) is a function from M into I. 1.2 Definition, [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 fuzzy singleton or fuzzy point 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 fuzzy 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 Definition, [3] Let A be a fuzzy 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 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]. 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 fuzzy set. 1.6 Definition, [4] Let f be a mapping from a set M into a set N, A be a fuzzy set in M and B be a fuzzy 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 fuzzy set in M defined by: f – 1 (B)(x)=B(f(x)), for all xM. 1.7 Definition, [5] Let f be a mapping 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 Definition, [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 Definition [7] 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)}. 1.11 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. 1.12 Definition [6] Let X and A be two fuzzy modules of an R-module M. A is called a fuzzy submodule of X if AX. 1.13 Proposition [7] Let A be a fuzzy set of 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 M such that A(x)X(x), xM. 1.14 Definition [8] A fuzzy module X of an R-module M is called fuzzy simple if and only if X has no fuzzy proper submodules. 1.15 Definition [8] A fuzzy module X of an R-module M is called fuzzy cyclic module, if there exists xtX such that each ykX written as y k=rℓxt for some fuzzy 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 Definition [6] Let X and Y be two fuzzy modules of R-modules M 1 and M 2 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. 1.17 Remark [9] 1- Let M and M' be two R-modules, f:MM' be an epimorphism. If A is a fuzzy submodule of M, then f(A) is a fuzzy submodule of M'. 2- Let M and M' be two R-modules, f:MM' be a homomorphism. If B is a fuzzy submodule of M', then f – 1(B) is a fuzzy submodule of M. 1.18 Definition [2] Suppose A and B be two fuzzy modules of R-module M. We define (A:B) by:- (A:B)={rt:rt is a fuzzy 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 Definition [10] Let X and Y be two fuzzy modules of M 1, M 2 respectively. Define XY:M1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 Proposition [10] Let X and Y are fuzzy modules of M 1 and M2 respectively, then XY is a fuzzy module of M 1M 2. 1.21 Proposition [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 Definition, [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 Definition Let X be a fuzzy module of an R-module M then X is called a chained fuzzy module if for each fuzzy submodules of X either A  B or B  A. To prove our next theorem, first we prove the following lemma: 2.3 Lemma Let A and B be two fuzzy subset of R then 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 Theorem A fuzzy 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 prove 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 fuzzy 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 prove 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=Z8 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 fuzzy submodules of Y then A, B are fuzzy submodules of X, since X is chained fuzzy module. Then AB or BA which implies Y is a chained fuzzy module. 2.7 Definition, [10] A fuzzy module X is called uniform fuzzy module if AB01 for any nontrivial fuzzy submodules A and B of X. 2.8 Proposition A fuzzy 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 prove 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 prove 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 the following proposition: 2.9 Proposition 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 then AB=B which implies AB=01. 2.10 Remark The converse of proposition (2.9) is not true for the following example 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 proposition (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 comparable 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 Definition Let A, B be two fuzzy submodules of a fuzzy module X of an R-module M, then A and B are called comparable if AB and BA. 2.13 Proposition A fuzzy module X of an R-module M is chained iff every two cyclic fuzzy submodules of X are comparable. Proof: Let A and B be fuzzy submodules of X. Suppose A  B, we show BA since A  B, there exists xtA and xtB A and  B. Let yk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). Thus BA so X is chained. The converse is obvious. 2.14 Remark A chained fuzzy module is indecomposable. Proof: Suppose X is decomposable, then X=AB for some fuzzy submodule A and B of X. Thus AB=O1 is a contradiction (proposition 2.9). Now, we introduce the notion of chained fuzzy ring. First we have the following definition. 2.15 Definition, [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 Definition A fuzzy 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 fuzzy 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 Definition, [8] A fuzzy module X of an R-module M is called multiplication fuzzy module if for each nonempty fuzzy submodule A of X, there exists a fuzzy ideal I of R such that A=IX. 2.19 Proposition 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. Thus AB or BA. 2.20 Definition Let X be a chained fuzzy module of an R-module M and let V(X)={(O1:Xt)XtX}. 2.21 Definition, [10] Let X be a non empty fuzzy module of R-module M. The fuzzy annihilator of A denoted by (F-annA) is defined by {Xt:xR,XtAO1},t[0,1], where A is a proper fuzzy submodule of X. Note that: (F-annA)(a)=sup{t:t[0,1],atAO1}, for all aR; that is F-annA=(O1:A). 2.22 Definition, [10] A fuzzy 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 then rℓX=O1 a contradiction. 2.24 Definition, [13] A fuzzy ideal A of a ring R is called fuzzy prime ideal, if A is non-constant and for any fuzzy ideals B and C of R such that B CA, then either BA or CA. Equivalently, A fuzzy ideal A of a ring R is called fuzzy prime 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 fuzzy prime ideal of R. Proof: (1) Let A, B  V(X) then A=(O1:xt) and B=(O1:yt) for some xtO1, yt=O1 and xt, ytX since X is chained fuzzy module then Xt is chained module (by theorem (2.4)) implies 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 fuzzy ideal of R. To show that P is a fuzzy prime ideal, let aℓ, bhR such that aℓbhP, then there is O1 xtX such that aℓbh(O1:xt). Then aℓbh xt=O1. This implies 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. Direct Sum of Chained Fuzzy Module We turn attention to the direct sum of chained fuzzy modules. 3.1 Remark If X and Y are two chained fuzzy modules of an R-module M1 and M 2 respectively then XY is not necessary chained fuzzy module of M1  M 2 as the following example shows: 3.2 Example Let X:Z6{0, 1 3 } such that 1 if a 2 X(a) 3 0 if a , 2          Let Y:Z6{0, 1 3 } such that 1 if a 3 Y(a) 3 0 if 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 fuzzy submodules of XY, where 1 if (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 Theorem Let X and Y be a fuzzy modules of an R-modules M 1 and M2 respectively, if XY is a chained fuzzy module of M 1M 2 then X is a chained fuzzy module of M 1 and Y is a chained fuzzy module of M2. Proof: By similar proof of theorem (4.10) in [14]. Next, we shall indicate the behaviors of chained fuzzy modules under homomorphism. 3.4 Theorem Let X and Y be a fuzzy modules of an R-modules M 1 and M2 respectively. Let f:XY be a fuzzy epimorphism. If X is a chained fuzzy module, then Y is a chained fuzzy 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), then BA. Therefore Y is a chained fuzzy module. 3.5 Proposition Let X and Y be two fuzzy modules of an R-modules M 1 and M 2 respectively. Let f:XY be a fuzzy homomorphism 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), then BA. Therefore X is a chained fuzzy module. 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. Mashinchi, M. and Zahedi, M. M., “On L-Fuzzy Primary Submodules”, Fuzzy Sets and Systems,.49:, pp.231-236, (1996). 4. Zadehi, M. 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Mukhrjee, T.K., (1989), Prime Fuzzy Ideals in Rings, Fuzzy Sets and Systems, 32:, 337-341. 14. Hadi, I.M.A., (2002), Some Types of Fuzzy Rings, Mathematics and Physics 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