ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or 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 Unive rsity of Baghdad Received in: 19 June 2011, Accepte d in: 20 September 2011 Abstract In this p aper we introduce the notion of semip rime fuzzy module as a generalization of semip rime module. We invest igate several characterizations and prop erties of this concept. Key Words: Prime fuzzy module, semiprime module, semip rime fuzz y 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 M ashinch and Zahedi [3]. The concept of fuzzy ideal of a ring by Liu in [4]. Dauns in [5] introduced the notion of semip rime submodules as a generalization of semip rime ideals of a ring. Eman in [6] st udied semip rime submodules. I.M .Hadi in [7] introduced the notion of semip rime fuzzy ideals of a ring also introduced semip rime fuzzy submodules of fuzzy module in [8]. Frias in [9] st udies semip rime module. In this p aper we introduce the notion of semip rime fuzzy modules as a generalization of R-semiprime modules and give many p rop erties of this concept. Throughout this p aper 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 p ap er. 1.1 De fini tion: [1] Let S be a non-emp ty set. A fuzzy set A in S (a fuzzy subset of S) is a function from S into [0,1]. 1.2 De finition: [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 Proposi tion: [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]. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 1.4 De finition: [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 De fini tion: [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 p rop erties of level subsets hold for each t  [0,1]. 1- (AB)t = At  Bt. 2- A = B iff At = Bt. 1.7 De finition: [1] Let f be a map p ing from a set M into a set N, let A be a fuzzy set in M and B be a fuzz y set in N. The image of A denoted by f(A) is t he fuzz y 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 t he 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 . 1.8 De fini tion: [2] Let M be an R-module. A fuzzy set X of M is called fuzz y 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 De fini tion: [5] Let X and A be two fuz zy modules of R-module M . A is called a fuzz y submodule of X if A  X. 1.10 Proposi tion: [6] Let A be a fuzz y set of an R-module M . Then the level subset At, t  [0,1] is a submodule of M iff A is a fuzz y 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-modu le M and xt  X t hen for all fuzz y singleton r k of R, rk xt = (r x), wher e  = min{k,t}. 1.12 De finition: [7] A fuzzy subset K of a ring R is called a fuzzy ideal of R if for each x, y  R: ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or 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 Proposi tion: [7] A fuzzy subset K of a ring R is a fuzz y ideal iff Kt, t [0,1] is an ideal of R. 1.14 De finition: [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 The orem: [2] Let A and B b e 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 De finition: [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 De finition: [9] Let X and Y be two fuzzy modules of M 1 and M 2 resp ectively defined XY:M 1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 Proposi tion: [9] Let X and Y be fuzz y modules of M 1 and M 2 resp ectively then XY is a fuzzy module of M 1M 2. 1.19 Remark: [9] Let A and B b e two fuzzy submodules of a fuzzy module X such that X = AB, then Xs = AsBs, for all s  [0,1]. 1.20 De finition: [9] A fuzzy module X of an R-module M is called a p rime fuzzy module if F-annA = F-annX, for any nontrivial fuzzy submodule A of X. 2- Semiprime Fuzzy Module s Firas in [9] introduced the concept of semip rime R-module (where M is called a semip rime module if for each r  R, x  M , r 2 x  M imp lies rx  M .. We shall fuzzify this concept in definition 2.3. But first we give the two definitions: 2.1 De fini tion: [7] Let A be a non constant fuzzy ideal of a ring R. A is called semiprime fuzz y ideal if for any fuzzy singleton xt  R, 2 t x  A, implies xt  A. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 2.2 De fini tion: [8] Let A be a fuzzy submodule of a fuzz y module X of an R-module M such that A  X, A is called semip rime fuzzy submodule if for each fuzzy singletone rt  R, xs  X, 2 t r xs  A imp lies rtxs  A. 2.3 De fini tion: Let X be a fuzzy module of an R-module M , X is called semip rime fuzzy module if for each non-zero fuzzy submodule A of X, F-annA is a semip rime fuzz y ideal of R. 2.4 Remarks: 1- Every p rime fuzz y module X is a semiprime fuzz y module. Proof: Let A be a fuzzy submodule of X. Since X is p rime, hence F-annA is a p rime fuzzy ideal by [17] which imp lies F-annA is a semip rime fuzz y ideal by [7]. Thus X is a semiprime fuzzy module. 2- If X is a semip rime fuzz y module, then F-annX is a semip rime fuzz y ideal. Proof: It is clear by definition 2.3, so is omitt ed. The following is a characterization of semiprime fuzz y module. 2.5 Proposi tion: Let X be a fuzzy module of an R-module M . Then X is a semiprime fuzz y module if and only if Xt is a semiprime module,  t  [0,1]. Proof: () Let N  Xt, t  [0,1]. To p rove annRN is a semip rime ideal of R. Let a 2 annRN  Xt. Since a 2 annRN, then a 2 N = 0, let x  N, hence a 2 x = 0. Assume X(x) = k. Hence xk  X, so  X. But F-ann is a semip rime fuzzy ideal. and 2 ka xk = (a 2 x)k  Ok  O1. Thus 2 ka  F-ann. Since F-ann is a semiprime fuzz y ideal. Thus akF-ann, hence akxk  O1, so (ax)k = Ok  O1. Thus ax = 0, for any x  N. () Conversely , to p rove F-annA is a semip rime fuzzy ideal of R for each non zero fuzzy submodule A of X. Let 2kr F-annA, so 2 kr xt  O1, for all xt  A. This imp lies (r 2 x) = 0, where  = min{k,t } hence r 2 x = 0, x  At. But At  Xt and Xt is semip rime by hy p othesis. Hence rx = 0. This imp lies (rx)  O1. That is rkxt  O1. Therefore rk  F-annA. By def. (2.3) we get t he result. The following p rop osition gives another characterization of semiprime fuzz y module. 2.6 Proposi tion: Let X be a fuzzy module of an R-module M . Then X is a semiprime fuzz y module if and only if O1 is a semiprime fuzz y submodule of X. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Proof: () Let 2tr xk  O1, for any fuzzy singleton rt of R, xk  X, hence 2 tr  (O1:xk) = F-ann. But F-ann is a semip rime fuzz y ideal. Thus rt  (O1:xk), so rtxk  O1. Thus O1 is a semiprime fuzz y submodule. () To p rove X is a semiprime fuzz y module. By p rop osition 2.5. It is enough to show that Xt is semip rime,  t  [0,1]. (i.e.) to p rove {0} is semip rime submodule of Xt by [9,Th.4.1.8]. Let r 2 = 0, to p rove r = 0, hence 2tr = Ot  O1, (rt) 2  O1, hence 2 tr  O1, which imp lies rt  O1, since O1 is semip rime. Thus r = 0, hence Xt is a semiprime module,  t  [0,1]. Thus X is a semip rime fuzz y module. Nex t we can give some examp les of semiprime and not semip rime fuzz y module. 2.7 Examples: 1- Let X:Z6  [0,1] defined by X(a) = 1,  a  Z6. Xt = Z 6, which is a semip rime module,  t  [0,1]. Thus by p rop . (2.5),X is a semip rime fuzz y module. 2- Let X:Z6  [0,1] defined by 1 if x {0, 3} X(x ) 1 otherwise 2        X0 = Z6, 1 2 X =Z 6 which is semip rime and  t > 1 2 , t X = {0, 3} is a p rime submodule, hence it is semip rime. Thus Xt is a semip rime 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 = Z 12, is not semip rime module. By p rop .(2.5) X is not semip rime fuzz y 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 semip rime fuzz y ideal of R, then [A:B] is a semip rime fuzz y ideal of R. Proof: Let 2ka  [A:B], hence 2 ka B  A. This imp lies 2 ka xt  A, for all xt  B. Hence 2 ka [A:] which is a semip rime fuzzy ideal. Thus ak  [A:]. So akxt  A, hence akB  A. T hus [A:B] is a semip rime fuzz y ideal. 2.9 Proposi tion: Let X be a fuzz y module of an R-module M . Then the following are equivalent: ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 1- X is semiprime. 2- [F-annA:B] is a semip rime fuzzy ideal of R for every nonzero fuzzy submodule A of X and for every non-zero fuzz y ideal B of R such that F-annA  F-annB. 3- [F-annA:xt] is a semip rime 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 fuzz y ideal of R for non-zero fuzz y singleton xk  X. Proof: (1)  (2) Let 2kr  [F-annA:B], hence 2 kr B  F-annA. So 2 kr bs  F-annA, for all bs B. Hence 2 2k sr b  F-annA, so 2(rb)  F-annA, where  = min{k,s}. Thus (rb)  F-annA, since F-annA is a semip rime fuzzy ideal. So rkbs  F-annA. Hence rk  [F-annA:B]. Thus [F-annA:B] is a semiprime fuzz y ideal. (2)  (3) It is followed by p utting = B. (3)  (4) It is easy to check that [F-annxt:<11>] = F-ann. But [F-ann:<11>] is a semip rime fuzz y ideal by (3). Thus F-annxt is a semiprime fuzz y ideal. The following p rop osition shows that the direct sum of semip rime fuzzy modules is semip rime fuzzy module. 2.10 Proposi tion: Let X and Y be two fuzzy modules of M 1 and M 2 R-modules resp ectively. Then X and Y are semip rime if and only if XY is a semiprime fuzz y module. Proof: () If X and Y are semip rime, then Xt and Yt are semip rime modules by p rop osition 2.5. Hence XtYt is a semip rime module by [9,p rop .4.1.11]. But XtYt = (XY)t by remark 1.19. Thus XY is a semiprime fuzz y module by p rop osition 2.5. () The proof is similarly. Now we t urn our at t ent ion t o image and inverse image of semiprime fuzzy module. W e h ave t he follo wing: 2.11 Pro pos i ti on : Let X and Y be t wo fuzzy modules of R-mo dules M1 an d M2 respect ively. Let f:M1  M2 be R- homo mo rp hism, t hen 3- Semiprime Fuzzy Module s and Other Relate d Fuzzy Module s In this section we st udy the relationship between semip rime fuzz y module and divisible, uniform and F-regular fuzzy modules. 3.1 De fi n i ti on : [1 7] A fuz zy module X is divisible if rtX = X, for all rt  Ot (rt is a fuzz y singleton of R). 3.2 De fini tion: [17] A fuzzy module is called uniform if AB  O1, for any non trivial fuzzy submodules A and B. 3.3 De fini tion: [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. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 3.4 Propos i ti on : Let X be a uniform fuzzy module. T hen X is a prime fuzzy module if an d only if X is semiprime fuzzy module. Pro of : () It is easy by p ro p.2.2. () T o pro ve F-an nX = F-an nA, fo r any non t rivial fuzzy submodule A of X [1 7,Def.3.1.1]. It is clear that F-annX  F-annA. T o p rove 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 y s such that y s  O1. Thus y s = aℓrtxk, aℓ is a fuzzy singleton of R. O1 = rty s = aℓ 2 tr xk, it follows 2 tr  F-ann since aℓxk  O1, so F-ann is a semip rime fuzzy ideal of R. Therefore rt  F-ann. This imp lies that O1 = rtaℓxk = y s. Thus y s = O1 which is a contradiction. 3.5 Proposi tion: 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 imp lies 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 Proposi tion: Let X be a uniform and semip rime fuzzy module. Then Xt is a quasi-Dedekind module,  t  (0,1]. Proof: X is uniform, imp lies Xt is uniform  t  (0,1] and X is semip rime, then Xt is semip rime by p rop . 2.5. Thus Xt is a quasi-Dedekind module  t  (0,1] by [9,p rop .2.4]. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 3.7 Proposi tion: Let X be a fuzzy module of an R-module M such that every fuzzy submodule of X is divisible, then X is semip rime. Proof: Let O1  xt  X. It is enough to show that F-ann is a semiprime fuzz y ideal by p rop .(2.9)(4). Let 2kr  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 semip rime fuzz y ideal. Thus X is semip rime fuzz y module. 3.8 Corollary: Let X be a fuzz y module and every fuzzy submodule of X is divisible. Then X is a p rime fuzzy module. Proof: By p rop .3.8, X is semiprime. But X is divisible, then X is p rime by [17,p rop .3.1.13]. 3.9 Proposi tion: Let X be an F-regular fuzzy module of an R-module M ,where R is a p rinciple ideal domain. Then X is a semip rime fuzz y 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,p rop .4.2.6]. Thus X is a semiprime fuzz y module. Re ferences 1. 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(2008), S-Compactly Packed Submodules and Semi-Prime M odules, M .Sc. Thesis, T ikrit University . 10. Zahedi, M .M . (1992), On L-Fuz zy Residual Quotient M odules and P.Primary Submodules, Fuz zy Sets and Sy st ems, 51:33-344. 11. Hamil, M .A. (2002), F-regular Fuz zy M odules, M .Sc. Thesis, Univ. of Baghdad. 12. Zahedi, M .M . (1991), Characterization of L-Fuzzy Prime Ideals, Fuz zy Sets and Fuz zy Sy st em, 44:147-160. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 13. M artinex, L. (1996), Fuzzy M odules Over Fuzzy Rings in Connection with Fuzzy Ideal of Ring, J. Fuzzy M ath., 4: 843-857. 14. M ukhegee, T.K.; Sen, M ,K, and Roy , D., (1996), On Submodules and Their Radicals, J. Fuz zy M ath., 4, p p .549-558. 15. Kumar, R., (1992), Fuz zy Cosets and Some Fuz zy Radicals, Fuz zy Sets and Sy st ems, 46: 261-265. 16. M ajumdar, S. (1990), Theory of Fuz zy M odules, Eull. Col. M ath. Sce., 82:395-399. 17. Rabi, H.J. (2001), Prime Fuz zy Submodules and Prime Fuz zy M odules, M .Sc.Thesis, Univ. of Baghdad. 18. Ali, S.M ijbass, (1997), Quasi-Dedekind M odules, Ph.D. Thesis, College of Science, Univ. of Baghdad. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or 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