مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Semiprime Fuzzy Modules M. A.Hamil Department of Mathematics , College of Education -Ibn-Al-Haitham University of Baghdad Received in: 19 June 2011, Accepted in: 20 September 2011 Abstract In this paper we introduce the notion of semiprime fuzzy module as a generalization of semiprime module. We investigate several characterizations and properties of this concept. Key Words: Prime fuzzy module, semiprime module, semiprime fuzzy module. Introduction The notion of fuzzy subsets of a set S  as a function from S into [0,1] was first developed by Zadeh [1]. The concept of fuzzy modules was introduced by Negoita and Ralescu in [2]. The concept of fuzzy submodule was introduced by Mashinch and Zahedi [3]. The concept of fuzzy ideal of a ring by Liu in [4]. Dauns in [5] introduced the notion of semiprime submodules as a generalization of semiprime ideals of a ring. Eman in [6] studied semiprime submodules. I.M.Hadi in [7] introduced the notion of semiprime fuzzy ideals of a ring also introduced semiprime fuzzy submodules of fuzzy module in [8]. Frias in [9] studies semiprime module. In this paper we introduce the notion of semiprime fuzzy modules as a generalization of R-semiprime modules and give many properties of this concept. Throughout this paper R is commutative ring with unity , M is an R-module and X is a fuzzy module of an R-module M. 1- Preliminaries In this section, we shall formulate the preliminary definitions and results that are required later in this paper. 1.1 Definition: [1] Let S be a non-empty set. A fuzzy set A in S (a fuzzy subset of S) is a function from S into [0,1]. 1.2 Definition: [2] Let xt:S [0,1] be a fuzzy set in S, where xS, t[0,1] defined by: t t if x y x (y) 0 if x y     for all y  S. xt is called a fuzzy singleton. 1.3 Proposition: [3] Let at, bk be two fuzzy singletons of a set S. If at = bk, then a = b and t = k, where t, k  [0,1]. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 1.4 Definition: [4] Let A and B be two fuzzy sets in S, then: 1- A = B iff A(x) = B(x), for all x  S. 2- A  B iff A(x)  B(x), for all x  S. 3- (AB)(x) = min{A(x),B(x)}, for all x  S, [2]. 1.5 Definition: [5] Let A be any fuzzy set in S for all t  [0,1], the set At = {x  S, A(x)  t} is called a level subset of A. 1.6 Remark: [1] The following properties of level subsets hold for each t  [0,1]. 1- (AB)t = At  Bt. 2- A = B iff At = Bt. 1.7 Definition: [1] 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 fuzzy set in N defined by: 1 1sup{A(z): z f (y)} if f (y) for all y N, f(A)(y) 0 otherwise         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. 1.8 Definition: [2] Let M be an R-module. A fuzzy set X of M is called fuzzy module of an R-module 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: [5] Let X and A be two fuzzy modules of R-module M. A is called a fuzzy submodule of X if A  X. 1.10 Proposition: [6] Let A be a fuzzy set of an R-module M. Then the level subset At, t  [0,1] is a submodule of M iff A is a fuzzy submodule of X, where X is a fuzzy module of an R-module M. 1.11 Remark: [5] If X is a fuzzy module of an R-module M and xt  X then for all fuzzy singleton rk of R, rk xt = (rx), where  = min{k,t}. 1.12 Definition: [7] A fuzzy subset K of a ring R is called a fuzzy ideal of R if for each x, y  R: مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 1- K(x – y)  min{K(x),K(y)}. 2- K(xy)  max{K(x),K(y)}. 1.13 Proposition: [7] A fuzzy subset K of a ring R is a fuzzy ideal iff Kt, t [0,1] is an ideal of R. 1.14 Definition: [2] Let A and B be two fuzzy submodules of a fuzzy module X of an R-module M. The residual quotient A and B denoted by (A:B) is the fuzzy subset of R defined by: (A:B)(r) = sup{t[0,1], rtB  A} for all r  R. That is (A:B) ={rt : rtB  A, rt is fuzzy singleton of R}. 1.15 Theorem: [2] Let A and B be two fuzzy submodules of a fuzzy module X of an R-module M. Then the residual quotient (A:B) of A and B is a fuzzy ideal of R. 1.16 Definition: [8] Let A be fuzzy submodule of a fuzzy module X. The fuzzy annihilator of A denoted by F-annA is defined by: (F-annA)(r) = sup{t : t[0,1], rtA  O1} for all r  R. That is F-annA = (O1:A). 1.17 Definition: [9] Let X and Y be two fuzzy modules of M 1 and M 2 respectively defined XY:M1M 2  [0,1] by (XY)(a,b) = min{X(a),Y(b)} for all (a,b)  XY. XY is called a fuzzy external direct sum of X and Y. 1.18 Proposition: [9] Let X and Y be fuzzy modules of M 1 and M2 respectively then XY is a fuzzy module of M 1M 2. 1.19 Remark: [9] 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]. 1.20 Definition: [9] A fuzzy module X of an R-module M is called a prime fuzzy module if F-annA = F-annX, for any nontrivial fuzzy submodule A of X. 2- Semiprime Fuzzy Modules Firas in [9] introduced the concept of semiprime R-module (where M is called a semiprime module if for each r  R, x  M, r2x  M implies rx  M.. We shall fuzzify this concept in definition 2.3. But first we give the two definitions: 2.1 Definition: [7] Let A be a non constant fuzzy ideal of a ring R. A is called semiprime fuzzy ideal if for any fuzzy singleton xt  R, 2 t x  A, implies xt  A. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 2.2 Definition: [8] Let A be a fuzzy submodule of a fuzzy module X of an R-module M such that A  X, A is called semiprime fuzzy submodule if for each fuzzy singletone rt  R, xs  X, 2 t r xs  A implies rtxs  A. 2.3 Definition: Let X be a fuzzy module of an R-module M, X is called semiprime fuzzy module if for each non-zero fuzzy submodule A of X, F-annA is a semiprime fuzzy ideal of R. 2.4 Remarks: 1- Every prime fuzzy module X is a semiprime fuzzy module. Proof: Let A be a fuzzy submodule of X. Since X is prime, hence F-annA is a prime fuzzy ideal by [17] which implies F-annA is a semiprime fuzzy ideal by [7]. Thus X is a semiprime fuzzy module. 2- If X is a semiprime fuzzy module, then F-annX is a semiprime fuzzy ideal. Proof: It is clear by definition 2.3, so is omitted. The following is a characterization of semiprime fuzzy module. 2.5 Proposition: Let X be a fuzzy module of an R-module M. Then X is a semiprime fuzzy module if and only if Xt is a semiprime module,  t  [0,1]. Proof: () Let N  Xt, t  [0,1]. To prove annRN is a semiprime ideal of R. Let a2annRN  Xt. Since a2annRN, then a2N = 0, let x  N, hence a2x = 0. Assume X(x) = k. Hence xk  X, so  X. But F-ann is a semiprime fuzzy ideal. and 2 ka xk = (a2x)k  Ok  O1. Thus 2 ka  F-ann. Since F-ann is a semiprime fuzzy ideal. Thus akF-ann, hence akxk  O1, so (ax)k = Ok  O1. Thus ax = 0, for any x  N. () Conversely, to prove F-annA is a semiprime fuzzy ideal of R for each non zero fuzzy submodule A of X. Let 2 kr F-annA, so 2 kr xt  O1, for all xt  A. This implies (r 2 x) = 0, where  = min{k,t} hence r2x = 0, x  At. But At  Xt and Xt is semiprime by hypothesis. Hence rx = 0. This implies (rx)  O1. That is rkxt  O1. Therefore rk  F-annA. By def. (2.3) we get the result. The following proposition gives another characterization of semiprime fuzzy module. 2.6 Proposition: Let X be a fuzzy module of an R-module M. Then X is a semiprime fuzzy module if and only if O1 is a semiprime fuzzy submodule of X. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Proof: () Let 2 tr xk  O1, for any fuzzy singleton rt of R, xk  X, hence 2 tr  (O1:xk) = F-ann. But F-ann is a semiprime fuzzy ideal. Thus rt  (O1:xk), so rtxk  O1. Thus O1 is a semiprime fuzzy submodule. () To prove X is a semiprime fuzzy module. By proposition 2.5. It is enough to show that Xt is semiprime,  t  [0,1]. (i.e.) to prove {0} is semiprime submodule of Xt by [9,Th.4.1.8]. Let r2 = 0, to prove r = 0, hence 2 tr = Ot  O1, (rt) 2  O1, hence 2 tr  O1, which implies rt  O1, since O1 is semiprime. Thus r = 0, hence Xt is a semiprime module,  t  [0,1]. Thus X is a semiprime fuzzy module. Next we can give some examples of semiprime and not semiprime fuzzy module. 2.7 Examples: 1- Let X:Z6  [0,1] defined by X(a) = 1,  a  Z6. Xt = Z6, which is a semiprime module,  t  [0,1]. Thus by prop. (2.5),X is a semiprime fuzzy module. 2- Let X:Z6  [0,1] defined by 1 if x {0, 3} X(x) 1 otherwise 2       X0 = Z6, 1 2 X =Z6 which is semiprime and  t > 1 2 , t X = {0, 3} is a prime submodule, hence it is semiprime. Thus Xt is a semiprime module,  t  [0,1]. Thus X is a semiprime fuzzy module. 3- Let X : Z12  [0,1] defined by: 1 if a {0,2, 4, 6, 8,10} X(a) 0 otherwise      It is clear that X is a fuzzy module and X0 = Z12, is not semiprime module. By prop.(2.5) X is not semiprime fuzzy module. 2.8 Lemma: Let A and B be two fuzzy submodules of fuzzy module of an R-module M. If for each xtB, [A:] is a semiprime fuzzy ideal of R, then [A:B] is a semiprime fuzzy ideal of R. Proof: Let 2 ka  [A:B], hence 2 ka B  A. This implies 2 ka xt  A, for all xt  B. Hence 2 ka [A:] which is a semiprime fuzzy ideal. Thus ak  [A:]. So akxt  A, hence akB  A. Thus [A:B] is a semiprime fuzzy ideal. 2.9 Proposition: Let X be a fuzzy module of an R-module M. Then the following are equivalent: مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 1- X is semiprime. 2- [F-annA:B] is a semiprime fuzzy ideal of R for every nonzero fuzzy submodule A of X and for every non-zero fuzzy ideal B of R such that F-annA  F-annB. 3- [F-annA:xt] is a semiprime fuzzy ideal of R for every non-zero fuzzy submodule A of X and for every fuzzy singleton xt  R such that xt  F-annA. 4- F-ann(xk) is a semiprime fuzzy ideal of R for non-zero fuzzy singleton xk  X. Proof: (1)  (2) Let 2 kr  [F-annA:B], hence 2 kr B  F-annA. So 2 kr bs  F-annA, for all bs B. Hence 2 2 k sr b  F-annA, so 2(rb)  F-annA, where  = min{k,s}. Thus (rb)  F-annA, since F-annA is a semiprime fuzzy ideal. So rkbs  F-annA. Hence rk  [F-annA:B]. Thus [F-annA:B] is a semiprime fuzzy ideal. (2)  (3) It is followed by putting = B. (3)  (4) It is easy to check that [F-annxt:<11>] = F-ann. But [F-ann:<11>] is a semiprime fuzzy ideal by (3). Thus F-annxt is a semiprime fuzzy ideal. The following proposition shows that the direct sum of semiprime fuzzy modules is semiprime fuzzy module. 2.10 Proposition: Let X and Y be two fuzzy modules of M 1 and M2 R-modules respectively. Then X and Y are semiprime if and only if XY is a semiprime fuzzy module. Proof: () If X and Y are semiprime, then Xt and Yt are semiprime modules by proposition 2.5. Hence XtYt is a semiprime module by [9,prop.4.1.11]. But XtYt = (XY)t by remark 1.19. Thus XY is a semiprime fuzzy module by proposition 2.5. () The proof is similarly. Now we turn our at tention to image and inverse image of semiprime fuzzy module. We have the following: 2.11 Proposition: Let X and Y be two fuzzy modules of R-modules M1 and M2 respectively. Let f:M1  M2 be R- homomorphism, then 3- Semiprime Fuzzy Modules and Other Related Fuzzy Modules In this section we study the relationship between semiprime fuzzy module and divisible, uniform and F-regular fuzzy modules. 3.1 De finition : [17] A fuzzy module X is divisible if rtX = X, for all rt  Ot (rt is a fuzzy singleton of R). 3.2 Definition: [17] A fuzzy module is called uniform if AB  O1, for any non trivial fuzzy submodules A and B. 3.3 Definition: [17] Let A be a fuzzy submodule of fuzzy module X. Then A is called an essential fuzzy submodule if AB  O1, for any nontrivial fuzzy submodule B of X. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 3.4 Proposition: Let X be a uniform fuzzy module. Then X is a prime fuzzy module if and only if X is semiprime fuzzy module. Proof: () It is easy by prop.2.2. () To prove F-annX = F-annA, for any non trivial fuzzy submodule A of X [17,Def.3.1.1]. It is clear that F-annX  F-annA. To prove F-annA  F-annX. Let rt  F-annA and rt  F-annX. Thus there exists xk  X, xk  0k such that rtxk ⊈ O1. Since X is uniform, A  O1, then there exists y s  A and ys such that ys  O1. Thus ys = aℓrtxk, aℓ is a fuzzy singleton of R. O1 = rtys = aℓ 2 tr xk, it follows 2 tr  F-ann since aℓxk  O1, so F-ann is a semiprime fuzzy ideal of R. Therefore rt  F-ann. This implies that O1 = rtaℓxk = y s. Thus y s = O1 which is a contradiction. 3.5 Proposition: If X is a uniform fuzzy module, then Xt is a uniform module,  t  (0,1]. Proof: Let N and W be submodules of Xt such that N  O, W  O. Define A: M  [0,1], B:M  [0,1] by t if x N, t 0 A(x) 0 otherwise      , t if x W, t 0 B(x) 0 otherwise      This implies A and B are fuzzy submodules of X and At = N, Bt = W, for all t  (0,1]. Since X is uniform, then AB  O1. But t if x N W (A B)(x) 0 if x N W        . On the other hand, (AB)t = At  Bt = N  W. Hence N  W  {0}. Thus Xt is a uniform module,  t  (0,1]. Recall that an R-submodule N of module M is called quasi-invertible if Hom( M N ,M) = 0. And an R-module M is called quasi-Dedekind if every non-zero R-submodule of M is quasi- invertible(18). 3.6 Proposition: Let X be a uniform and semiprime fuzzy module. Then Xt is a quasi-Dedekind module,  t  (0,1]. Proof: X is uniform, implies Xt is uniform  t  (0,1] and X is semiprime, then Xt is semiprime by prop. 2.5. Thus Xt is a quasi-Dedekind module  t  (0,1] by [9,prop.2.4]. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 3.7 Proposition: Let X be a fuzzy module of an R-module M such that every fuzzy submodule of X is divisible, then X is semiprime. Proof: Let O1  xt  X. It is enough to show that F-ann is a semiprime fuzzy ideal by prop.(2.9)(4). Let 2 kr  F-ann, hence 2 kr xt = O1. But is a divisible fuzzy submodule. Then = rk ; rk is a fuzzy singleton of R. Hence xt = rkcℓxt, cℓ  X. So rkxt = rkrkcℓxt = 2 kr cℓxt, but 2 kr cℓxt = cℓ 2 kr xt = cℓO1 = O1. Thus rkxt = O1. So rk  F-ann. Thus F-ann is a semiprime fuzzy ideal. Thus X is semiprime fuzzy module. 3.8 Corollary: Let X be a fuzzy module and every fuzzy submodule of X is divisible. Then X is a prime fuzzy module. Proof: By prop.3.8, X is semiprime. But X is divisible, then X is prime by [17,prop.3.1.13]. 3.9 Proposition: Let X be an F-regular fuzzy module of an R-module M ,where R is a principle ideal domain. Then X is a semiprime fuzzy module. Proof: Since X is F-regular, then Xt is F-regular  t  (0,1] by [11]. Hence Xt is a semiprime module  t  (0,1] by [9,prop.4.2.6]. Thus X is a semiprime fuzzy module. References 1. Zadeh, L.A., (1965), Fuzzy Sets, Information and Control, 8: 338-353. 2. Negoita, C.V. and Ralescu, D.A., (1975), Applications of Fuzzy Sets and System Analysis (Birkhous, Basel). 3. Maschinchi, M. and Zahedi, M.M., (1992), On L-Fuzzy Primary Submodules, Fuzzy Sets and Systems, 49:231-236. 4. Liu, W.J. (1982), Fuzzy Invariant Subgroups and Fuzzy Ideals, Fuzzy Sets and Systems, 8:133-139. 5. Dauns, (1980), Prime Modules and One-Sided Ideals in "Ring Theory and Algebra III" (Proceeding of the Trird Oklahama Conference), B.R. McDonald, NewYork, .301-314. 6. Athab, E.A. 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(1996), Fuzzy Modules Over Fuzzy Rings in Connection with Fuzzy Ideal of Ring, J. Fuzzy Math., 4: 843-857. 14. Mukhegee, T.K.; Sen, M,K, and Roy, D., (1996), On Submodules and Their Radicals, J. Fuzzy Math., 4, pp.549-558. 15. Kumar, R., (1992), Fuzzy Cosets and Some Fuzzy Radicals, Fuzzy Sets and Systems, 46: 261-265. 16. Majumdar, S. (1990), Theory of Fuzzy Modules, Eull. Col. Math. Sce., 82:395-399. 17. Rabi, H.J. (2001), Prime Fuzzy Submodules and Prime Fuzzy Modules, M.Sc.Thesis, Univ. of Baghdad. 18. Ali, S.Mijbass, (1997), Quasi-Dedekind Modules, Ph.D. Thesis, College of Science, Univ. of Baghdad. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 ة الضبابیةالمقاسات شبه األولی میسون عبد هامل جامعة بغداد، ابن الهیثم -كلیة التربیة ،قسم الریاضیات 2011 ایلول 20: ، قبل البحث في 2011 حزیران 19: استلم البحث في الخالصة م اعطیـت العدیـد مــن ثــ.ةالولیـ للمقاســات شـبه ا"اعمامـا ة الـضبابیةفـي هـذا البحـث قــدم مفهـوم المقاسـات شــبه االولیـ .التشخیصات والخواص لهذا المفهوم . المقاس األولي الضبابي ، المقاس شبه األولي ، المقاس شبه األولي الضبابي:الكلمات المفتاحیة