Conseguences of soil crude oil pollution on some wood properties of olive trees Mathematics | 192 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 Quasi-Fully Cancellation Fuzzy Modules Hatem Yahya Khalaf Department of Mathematics/ College of Education(Ibn-Al-Haitham), University of Baghdad Hadi Ghali Rashed Al-Rusafa the First Education/ Ministry of Education Received in 5/December /2016 Accepted in: 28/Decmber/2016 Abstract In this paper it was presented the idea quasi-fully cancellation fuzzy modules and we will denote it by Q-FCF(M), condition universalistic idea quasi-fully cancellation modules It .has been circulated to this idea quasi-max fully cancellation fuzzy modules and we will denote it by Q-MFCF(M). Lot of results and properties have been studied in this research. Key word : Q-FCF(M), Q-MFCF(M), Direct Sum Q-FCF(M). Mathematics | 193 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 Introduction Let R be a commutative ring with identity . Let X be a fuzzy module of an R-module M and we will denote it by (X be F(M)) . X is called Q-FCF(M) , if for every fuzzy ideal I of R and two every fuzzy submodules A and B of X , if IA=IB , then A+(F- annI)=B+(F-annI) where F-annI is a fuzzy submodule of X and define by F-annI ={xt X: Ixt=01} see Definition (1.1) . And X is called Q-MFCF(M) if for every maximal fuzzy ideal of R and for every two fuzzy submodules A and B of X such that IA=IB implies that A+(F-annxI)=B+(F-annxI) see (2.1). Clearly, every Q-FCF(M) is Q-MFCF(M) see (Remark(2.3)(1)) and every fully cancellation fuzzy module is Q-FCF(M). See (Remark(1.3)(1)) but the convers is not true ingenarl see (Remark(1.6)) And , if X is multiplication and naturally cancellation fuzzy module or (fuzzy Principle ideal ) , then X is Q-FCF(M). See Proposition (1.4) and Proposition (1.5 ). In this chapter , we will study in details the concept of Q-FCF(M).This chapter consists of three parts. In part one we give some basic propositions of Q-MFCF(M). It turns out that a fuzzy module X is quasi-fully cancellation (quasi-max fully cancellation) if and only if I IK, then H K+(F-annxI), where H and K are fuzzy submodules of X and I be a fuzzy ideal (fuzzy maximal ideal) of R. Equivalently , I(ht) IL, then ht L+(F-annxI), where ht X if and only if (IH:I)=H+(F-annxI) where L is a fuzzy submodule . And ht is a fuzzy singleton of R, see Proposition (1.7) and Proposition (2.9).And fully cancellation fuzzy module is equivalent quasi-fully cancellation in the class X is torsion free fuzzy module over a fuzzy integral domain R see Proposition (1.4) . Part two is devoted to study the relation between max-fully cancellation fuzzy module and Q-MFCF(M) but the convers is not true see Remark (2.7) and Example (2.8) . Part three is study the concepts the direct sum of Q-FCF(M) which is mentioned in chapter one section five. And naturally cancellation fuzzy module X is equivalent to quasi-fully cancellation if X is multiplication fuzzy module. § 1. Quasi-Fully Cancellation Fuzzy Modules In this part we give the concept of Q-FCF(M) this concept is generalization of concept quasi-fully cancellation modules[1], and we give a some basic results and properties of this concept. Also, relationships between the class of Q-FCF(M)and other types of modules are established . " Recall that an R-module M is called quasi-fully cancellation produle. If for every ideal I of R and for every two submodules A,B of M. Such that IA=IB implies A+annMI=B+annMI (where annMI={m M, Im=0}.[1] " Here ,we introduce the principle definition of our work. Definition 1.1: Let X beF(M). X is called Q-FCF(M) for every fuzzy ideal I of R and for every fuzzy submodules A and B of X , if IA=IB, then A+F-annXI=B+F-annXI Where F-annI is a fuzzy submodule of X. And definition by {xt X: I.xt=01}. t (0,1]. Proposition 1.2: Let X beF(M). such that (F-annX)t =F-annXt . Then X is a Q-FCF(M)if and only if Xt is a quasi-fully cancellation module, t (0,1]. Mathematics | 194 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 Proof: Let X beF(M). Let N and K be two submodules of M, and let J be an ideal of R. such that JN=JK. Now, define: A:M [0,1] , B: [0,1] by A(x)={ t (0,1] B(x)={ t (0,1] and Define : I:R [0,1] by I(x)={ t (0,1] It is clear that A and B are fuzzy submodules of X and I be a fuzzy ideal of R. Also , At=N, Bt=K and It=J ; t (0,1] , JN=JK, then ItAt= ItBt Therefore (IA)t=(IB)t t (0,1] . Thus IA=IB. ButX is a Q-FCF(M). Then we get A+F-annXI =B+F-annXI Hence (A+F-annXI)t=(B+F-annXI)t. t (0,1] But (F-annXI)t =F- annMIt t (0,1] see[5,Proposition.(2.2)] So, At+F-annIt=Bt+F-annIt. Therefore N+F-annJ=K+F-annJ. Thus Xt is quasi-fully cancellation module. Another side, let A,B be two fuzzy submodules of X and let I be a fuzzy ideal of R. such that IA=IB. To prove A+F-annI=B+F-annI. Now, since IA=IB, then (IA)t=(IB)t t (0,1], so , ItAt= ItBt But Xt is a quasi-fully cancellation module. Then At+F-annIt=Bt+F-annIt. But (F-annI)t=F-sannIt t (0,1] which implies At+(F-annI)t=Bt+(F-annI)t. So (A+F-annXI)t=(B+F-annXI)t. , by [2 , Remark (1.1.7)]. Thus A+F-annXI=B+F-annXI. Therefore X is Q-FCF(M). Remarks and Examples 1.3: (1) Every fully cancellation fuzzy module is Q-FCF(M). But the converse is not true ingeneral by the following example: Let M=Z4 is a Z-module . LetX : M [0,1] define by X(x)={ Let A: ( ̅ [0,1] define by A(x)={ ̅ t (0,1] Let B: Z4 [0,1] define by B(x)={ t (0,1] Define I: ( [0,1] define by I(x)={ t (0,1] It is clear that A and B are fuzzy submodules of X and I is a fuzzy ideal of R Mathematics | 195 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 M=Z4=Xt is not fully cancellation module by [3, Remark and Examples (2.3)(2)]. Since ( ( ̅ = ( Z4=0, but ( ̅ ≠Z4. Implies that X is not fully cancellation fuzzy module by [4, Proposition (1.2,2)]. Hence It=( , At=( ̅ and Bt=Z4 . So ItAt= ItBt (since ( ( ̅ =( Z4= ̅ Also, annIt=ann ( =Z4 Then At+annIt=( ̅ +Z4=Z4 Also , Bt+annIt=Z4+Z4=Z4 Then Xt=M is quasi-fully cancellation module. and by proposition (1.2) . X is Q-FCF(M). (2) Any fuzzy module of a Z-module Z is Q-FCF(M). Proof: By[4, Remark and Examples (1.2.3)(1)] we get X is fully cancellation fuzzy module. And by (1) we interduce X is Q-FCF(M). (3) Every fuzzy submodule of Q-FCF(M) is quasi-fully cancellation. Proof: Let X be a Q-FCF(M) of an R-module M. let N,K be two submodules of M and J be an ideal of R. let C be a fuzzy submodule of X . To prove C is Q-FCF(M). Define: C: M [0,1] by C(x)={ Define A: N [0,1] by A(x)={ t (0,1] Define B: [0,1] by B(x)={ t (0,1] Define I: J [0,1] by I(x)={ t (0,1] It is clear that A,B are fuzzy submodules of C and I is a fuzzy ideal of R. Also, It=J , At=N , Bt=K , Ct=M. Since X is Q-FCF(M). Then Xt is quasi-fully cancellation module.by Proposition (1.2). Since C is a fuzzy submodule of X. Then Ct is a submodule of Xt and by [ 3 , Remark and Examples (2.2)], we get Ct is quasi-fully cancellation module. Therefore C is Q-FCF(M). by Proposition(1.2) Mathematics | 196 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 (4) Let X1and X2 be two fuzzy modules of an R-module M1, M2 respectivly such that M1 M2.Then X1 is Q-FCF(M)if and only if X2 is Q-FCF(M). Proof: ( ) Let X1: M1 [0,1] define by X1(x)={ Let X2: M2 [0,1] define by X2(x)={ It is clear that X1 and X2 are fuzzy modules of M1 and M2 respectively . Since (X1)t=M1 , (X2)t=M2 and M1 M2 , t (0,1] , then M2 is quasi-fully cancellation module by [ 3 ,Remark and Examples (2.2). (5)] Then X2 is Q-FCF(M) by Proposition (1.2). Conversely: it is clear. Proposition 1.4: Let X be a multiplication and naturally cancellation fuzzy module of an R-module M then X is Q-FCF(M). Proof: Since X is multiplication and naturally cancellation fuzzy module, then by[4, Theorem (1.4.3)]. We obtian X is fully cancellation fuzzy module and by Remark and Examples( ( 1.3 ) (1)). X is Q-FCF(M). Proposition 1.5: Let X be a fuzzy torsion free module over a fuzzy integral domain R. If X is quasi-fully cancellation modules, then X is fully cancellation. Proof: Let X be a fuzzy torsion free module and R be a fuzzy integral domain. Suppose that IA=IB where A,B are two fuzzy submodules of X and I be a fuzzy ideal of R. Since X is quasi-fully cancellation, then A+(F-annxI)=B+(F-annxI). Now, let xt F-annxI, then Ixt=01 t (0,1] And hence xt=01 for each fuzzy singleton of I, (0,1]. ≠01 (since R is integral domain). Then I≠01. Therefore xt=01 (since X is a fuzzy torsion free). Then F-annxI=01.Thus A=B and hence X is a fully cancellation fuzzy module. Mathematics | 197 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 Proposition 1.6: Let X be F(M) and let R be a fuzzy principle ideal ring . Then X is a quasi-fully cancellation module. Proof: Let A and B be two fuzzy submodules of X. Let I be a fuzzy ideal of a fuzzy principle ideal ring R. Suppose that IA=IB. We show that A+(F-annxI)=B+(F-annxI) Since R is a fuzzy principle ideal ring, then I=( ), where be a fuzzy singleton of R, (0,1]. Then ( ) A=( ) B and hence .at= .bs where at A and bs B (0,1]. Now, .at- .bs=01 Then (ra-rb) =0 ≤ 01 where =min{ ,t ,s}. Therefore (a-b) =01,and hence (at-bs)=01 .Thus at-bs F-annxI. But at=bs+ at-bs B+(F-annxI) . Then A B+(F-annxI) Hence A+(F-annxI) B+(F-annxI) Similarly: B+(F-annxI) A+(F-annxI) Thus A+(F-annxI)= B+(F-annxI) And hence X is Q-FCF(M). Remark 1.7: The converse of Remark and examples( (1.3)(1)) is not true in general by the following example: Let M=Zp∞ and R=Z define by X:M [0,1] such that X(x) ={ It is clear that t (0,1], Xt=M and M is not fully cancellation module [3, Examples (2.5)(2)]. Thus X is not fully cancellation fuzzy module by Proposition ( 1.2) . Proposition 1.8: Let X be F(M) and let H,K and L are fuzzy submodules of X. let I be a fuzzy ideal of R. Then the following statements are equivalent:- 1- X is a Q-FCF(M). Mathematics | 198 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 2- If IH IK, then H K+(F-annxI). 3- I(ht) IL, then ht L+(F-annxI), where ht X. (0,1]. 4- (IH:xI)=H+(F-annxI). Proof: (1) (2) let X be a Q-FCF(M) and let IH IK. Then IK IH+IK=I(H+K) by [5,proposition (2.6)] . Hence K+(F-annxI)=(H+K)+(F-annxI) (since X is Q-FCF(M)). Therefore H K+(F-annxI). (2) (3) let I(ht) IL, where ht H and by (2) we have (ht) L+(F-annxI). Thus ht L+(F-annxI). (3) (4) let xt (IH:xI) , (0,1], then Ixt IH and by (3) xt H+(F-annxI). Therefore (IH:xI) H+(F-annxI). Conversely, let H+(F-annxI), (0,1]. Then =ht+ms , where ht H and ms (F-annxI), (0,1]. Thus I =Iht +Ims . But Ims=01 Therefore I =Iht IH . Then (IH:xI). Thus H+(F-annxI) (IH:xI). and hence (IH:xI)= H+(F-annxI). (4) (1) let IH=IK we want to prove X is Q-FCF(M). i.e To prove H+(F-annxI)=K+(F-annxI) K (IH:xI) and by (4) we get (IH:xI) = H+(F-annxI) . Hence K H+(F-annxI). Therefore K+(F-annxI) H+(F-annxI). Similarly: H (IK:xI) and by (4), we obtain(IK:xI)= K+(F-annxI), then H K+(F-annxI). Therefore H+(F-annxI)=K+(F-annxI). Thus X is Q-FCF(M). Proposition 1.9: Let X be F(M).Then X is Q-FCF(M) if and only if ((A+(F-annxI):B)=(IA:RIB) where I be a fuzzy ideal of R and A,B are two fuzzy submodules of X. Proof: ( ) Let xt ((A+(F-annxI):B) , (0,1]. Mathematics | 199 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 Then xtB (F-annxI) and hence xt bs A+(F-annxI) for and bs B, (0,1]. Thus xtIB IA . Then xt (IA:RIB). Therefore ((A+(F-annxI):B) (IA:RIB). Now, let xt (IA:RIB), then xtIB IA and by Proposition (1.7),we get xtB A+(F-annxI) and hence xt ((A+(F-annxI):B). Thus (IA:RIB) ((A+(F-annxI):B). Therefore ((A+(F-annxI):B)=(IA:RIB). On the other side: suppose IB IA, where I be a fuzzy ideal of R and A,B are two fuzzy submodules of X. Then (IA:RIB)=R. But ((A+(F-annxI):B)=(IA:RIB) . Then ((A+(F-annxI):B)=R, it follows that B A+(F-annI). Thus X is Q-FCF(M). § 2. Quasi-Max Fully Cancellation Fuzzy Modules As we have mentioned in section one, that every fully cancellation fuzzy module is Q- FCF(M) and the converse is not to be true in general. In this section, we introduce the concept of Q-MFCF(M) and to show that every max-fully cancellation fuzzy module is Q-FCF(M) but the converse is not true . Morevore , we prove that in the class of faithful fuzzy module , the two concepts max-fully cancellation fuzzy module and Q-MFCF(M) are equivalent. " Recall that an R-module M is called quasi-max fully cancellation module if for every maximal ideal I of R and for every two submodules N and K of M such that IN=IK implies N+annMI=K+annMI [6]." We shall fuzzify this concepts as follows: Definition 2.1 : Let X be F(M). X is called Q-MFCF(M) if for every maximal fuzzy ideal of R and for every two fuzzy submodules A and B of X such that IA=IB implies that A+(F-annxI)=B+(F- annxI). Next , we have the following Proposition. Proposition 2.2: Let X be F(M) and let I be a maximal fuzzy ideal of R. such that (F-annI)t=F-annIt , then X is a Q-MFCF(M) if and only if Xt is a quasi-max fully cancellation module t (0,1]. Proof: It is similar of proof of Proposition (1.2) only we take I maximal fuzzy ideal. Mathematics | 200 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 Remarks and examples 2.3:- (1) Every Q-FCF(M) is a Q-MFCF(M). Proof: it is clear. (2) A fuzzy module X of an Z6-module Z6 is Q-MFCF(M). Proof: Let M=Z6 and X :M [0,1] such that X(x) ={ Define: I:( ̅ [0,1] by I(r) ={ ̅ , t (0,1]. Define: A:( ̅ [0,1] by A(x) ={ ̅ Define: B:Z6 [0,1] by B(x) ={ It is clear that At=( ̅ , Bt=Z6 and It=( ̅ is a maximal ideal and Xt=M, t (0,1]. Now, ItAt=( ̅ ̅ = ( ̅ =ItBt=( ̅ Similarly if At=(0), Bt=( ̅ since ItAt=( ̅ ̅ = ( ̅ ̅ =ItBt=( ̅ where (0), ̅̅ ̅ ̅ are submodules of M=Z6 Then we obtain ̅ +annM ̅ = ̅ +annM ̅ = ̅ Thus M is Q-MFC(M).Therefore X is Q-MFCF(M). by Proposition (2.2). (3) The fuzzy module X of an Z4-module Z4 is a Q-FCF(M). Proof: By Remark and Examples ((1.3)(2)) . We have Z4 is quasi max fully cancellation module and by Proposition (2.2) we get the result. (4) Let X1 and X2 be a fuzzy modules of an R-module M1,M2 respectively . if M1 is a Q-MFC(M)and M1 M2 , then X1 is a Q-MFCF(M) if and only if X2 is a Q-MFCF(M). Proof: ( ) let X1 :M1 [0,1] define by X1(x) ={ Let X2 :M2 [0,1] define by X2(x) ={ Clear that (X1)t=M1 ,is a quasi-max fully cancellation module, then X1 is a Q-MFCF(M) by Proposition (2.7). But M1 M2 , then M1 is a quasi-max fully cancellation module by [6, Remark and Examples (1.3)(4)] Therefore X2 is a Q-MFCF(M). Mathematics | 201 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 Conversely: it is clear. (5) Let X be F(M).and let C be a fuzzy submodule of X, then C is Q-MFCF(M). Proof: The prove is similar of Remarks and Examples (1.3)(4) ) only we take the ideal of a ring R in maximal fuzzy ideal. The following Lemmas are needed to prove next Proposition. Lemma 2.4: Let X be F(M) and let A,B and C are fuzzy submodules of X such that C B.Then C+(B A)=(C+A) B. Proof: First to show that C+(B A) , (since C B and B A B) Then C+(B A) B by [2]. Further C C+A , B A C+A. Thus C+(B A) C+A Therefore C+(B A) B (C+A). Conversely: to show that C+ (B A) Let bt t (0,1]. Then bt= + for some fuzzy singletons C, A (0,1]. Hence bt=(c+a)t where t=min{ }. Then b=c+a by[7, Definition (1.1.3)(3)]. Hence a=b-c and =bt- Thus B (since bt B and C B ) Hence B A and so bt= + C+ (B A). Therefore C+(B A)=(C+A) B. Lemma 2.5: Let A be a fuzzy submodule of a fuzzy module X of an R-module M, let I be any fuzzy ideal of R, then F-annAI=F-annxI A. Proof: Since F-annAI is a fuzzy submodule of a fuzzy submodule A. then F-annxI is a fuzzy submodule of X . and F-annxI A F-annAI Conversely: to show that F-annAI F-annxI A? Mathematics | 202 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 Let xt F-annAI , where xt A , t (0,1]. Then xtI=01 01=01 A = xtI A=F-annAI A But xt A X xt X. Thus F-annxI A=01= xtI= F-annAI Now, we have the following Proposition. Proposition 2.6: Let A be a fuzzy submodule of a Q-MFCF(M) X, then A is a Q-MFCF(M). Proof: Let B,C are two fuzzy submodules of a fuzzy submodule A and let I be a maximal fuzzy ideal of R. such that IB=IC, then B,C are fuzzy submodules of X. Since X is a Q-MFCF(M) Then B+(F-annxI)=C+(F-annxI). But F-annAI=F-annxI A by Lemma (2.5). Hence B+ F-annAI=B+((F-annxI) A) =(B+(F-annxI)) A by Lemma (2.4). =(C+(F-annxI)) A =C+((F-annxI) A) by Lemma (2.4) =C+( F-annAI) by Lemma (2.5) Thus A is a Q-MFCF(M). Remark 2.7: Every max-fully cancellation fuzzy module is Q-MFCF(M). Proof: It is clear . The converse of Remark ( 2.7) is not true in general for example:- Let M=Z6, R=Z6 Let X :M [0,1] define by X(x) ={ Xt=M t (0,1], then X is a Q-MFCF(M) by Remark((1.3)(1)) and it is not max-fully cancellation fuzzy module since Let I:( ̅ [0,1] define by I(r) ={ ̅ , t (0,1]. And A:( ̅ [0,1] define by A(x) ={ ̅ Mathematics | 203 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 Also, B:( ̅ [0,1] define by B(x) ={ ̅ , t (0,1]. It is clear that A,B are fuzzy suodules of X and I is a fuzzy ideal of R. IA=IB ItAt=ItBt. ( ̅ ( ̅ ̅ ̅ ̅ , but ( ̅ ( ̅ Then At≠Bt A≠B. Thus X is not max-fully cancellation fuzzy module. The convers of Remark (2.7) is true under the following conditions. Proposition 2.8: Let X be F(M) and let F-annxI=01 be (F-faithful) for every non-empty fuzzy ideal I of R. Then every Q-MFCF(M) is max-fully cancellation fuzzy module. Proof: Let I be a maximal fuzzy ideal of R. and let A,B be two fuzzy submodules of X such that IA=IB Since X is Q-MFCF(M) , then A+(F-annxI)=B+(F-annxI). But (F-annxI)=01. Thus A=B Then X is max-fully cancellation fuzzy module. The following is characterization of Q-MFCF(M). Theorem 2.9: Let X be F(M), let A,B be two fuzzy submodules of X and let I be a maximal fuzzy ideal of R. Then the following statements are equivalent :- (1) X is a Q-MFCF(M). (2) If IA IB , then A B+(F-annxI). (3) If I(xt) IB, then xt B+(F-annxI).where xt X. (4) (IA:xI)= B+(F-annxI). Proof: Compare this proof with the proof of Proposition (1.7). The next result gives another characterization for Q-MFCF(M). Proposition 2.10: Let X be F(M).Then for any maximal fuzzy ideal I of R the following statements are equivalent:- (1) X is Q-MFCF(M). (2) For every fuzzy submodules A,B of X then ((A+(F-annxI)):B)=(IA:IB). Mathematics | 204 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 Proof: It is similar proof of Proposition (1.8). §3. The Direct Sum of Quasi-Fully Cancellation Fuzzy Modules and Its Generalization In this part we study the direct sum of two Q-FCF(M) and we prove some results about it. Also, we study its generalizations . First, we give the following lemma, which is needed in the next our Proposition. Lemma 3.1: Let X be F(M) , M=M1 M2 where M1, M2 are submodules of M, if X=A1 A2, where A1, A2 are fuzzy submodules of X , then F-annxI=F-annA1I F-annA2I where I is a fuzzy ideal of R. Proof: We must prove that F-annxI F-annA1I F-annA2I Let xt F-annxI, then Ixt=01 and xt X, t Since xt X x Xt=M= M1 M2. Then x=x1+x2 for some x1 M1, x2 M2 Define I: J [0,1] by I(x) ={ , t (0,1]. Be a fuzzy ideal of R. It is clear tat It=J Then J.x=J(x1+x2)=0 (since + on M is a direct sum) it follows That:- Jx1=Jx2=0. (Jx1)t=(Jx2)t=0t ≤ 01 t (0,1]. It(x1)t=It(x2)t=0t (since J=It). I(x1)t=I(x2)t=0t. Thus (x1)t F-annA1I and (x2)t F-annA2I. . Therefore xt=(x1)t+(x2)t F-annA1I F-annA2I . Conversely: let xt F-annA1I F-annA2I Then xt ((x1 ,(x2)s) where (x1 F-annA1I and (x2)s F-annA2I (0,1]. Thus I(x1 =01 and I(x2)s=01. Therefore I(x1 + I(x2)s=01, and hence I((x1 + (x2)s)=01 Then [J(x1+x2) = ≤ 01 where =min {t,s, } implies, J(x1+x2)=0 Therefore Jx=0 and hence I(xt)=01 Mathematics | 205 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 Hence xt F-annxI .Therefore F-annxI= F-annA1I F-annA2I Proposition 3.2: Let X be F(M) and let X= A1 A2, where A1 and A2 be two fuzzy submodules of X. such that F-annA1I F-annA2I= R where R(x)=1; x R. Then A1 and A2 are Q-FCF(M). Proof: ( ) Let I be a non-empty fuzzy ideal of R and let A and B are fuzzy submodules of X. Suppose that IA=IB we show that A+(F-annxI)=B+(F-annxI) Since (F-annA1)+ (F-annA2)= R , then by [4,Lemma (1.5.5)] we get , A= A1 A2 and B= B1 B2 for some fuzzy submodules A1 , A2 of A and for some fuzzy submodules B1, B2 of B. Now, I( A1 A2) =I( B1 B2) . Hence (IA1 , IA2)=(IB1 , IB2). [ 7 , Proposition (3.2.4)] Which implies that , IA1=IB1 and IA2=IB2 But A1 and A2 are Q-FCF(M). Thus A1+(F-annA1I)=B1+(F-annA1I) and A2+(F-annA2I)=B2+(F-annA2I) It follows that , A1+A2+(F-annA1I)+(F-annA2I)= B1+B2+(F-annA1I)+(FannA2I) Then we have : A+(F-annxI)=B+(F-annxI) Therefore X is Q-FCF(M). ( ) it is clear by used Remarks and Examples ((1.3) (4)). We end this section by the following result. Proposition 3.3: Let X be F(M) and let X= A1 A2 where A1 and A2 are two fuzzy submodules of X, such that F-annA1+ F-annA2= R where R(x)=1, x R. Then A1 and A2 are Q-MFCF(M)if and only if X is Q-MFCF(M). Proof: First side, let A and B be two fuzzy submodules of X and let I be a maximal fuzzy ideal of R. Since F-annA1+F-annA2= R, where R(x)=1, x R.then by[4, Lemma (1.5.5)] we get , A= A1 A2 and B= B1 B2, and by similar procedure as in the Proposition ( 3.2 ) . the required result can be obtained . Another side, it is clear by Remarks and Examples( (2.3)(5)). Mathematics | 206 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 References 1. Inaam,M.A.Hadi, Alaa,A.Elewi (2014)"quasi-fully cancellation module" Iraqi Journal of science (2014), Vol55,No.3A pp(1080-1085). 2. Gada, A.A., (2000),"Fuzzy Spectrum of a Modules Over Commutative Ring",M.Sc.Thesis , University of Baghdad . 3. Inaam,M.A.Hadi, Alaa, A. Elewi( 2014), “Fully cancellation and naturally cancellation module” journal of Al-Nahrain university , vol. 17(3).sep, pp.178-184. 4. .Hatam,Y.Khalaf., and Hadi,G.Rashed(2016),"Fully and Naturally Cancellation Fuzzy Modules"International Journal of Applied Mathematics statistical Sciences (IJAMSS):Vol.(5).pp.2319-3980. 5. Inaam ,M.A.Hadi, Maysoun, A. Hamil. (2011),”Cancellation and Weakly Cancellation Fuzzy Modules” Journal of Basrah Reserchs((sciences)) Vol.37 No.4.D . 6. Bothaynah,N.Shihab and Heba,M.A Jude (2015)” Max-fully cancellation modules” Journal of Advances In Mathematics. Vol.11No.7,p5462-5475 7. Layla,S.M.And Shrooq,B.S.(2003),"Semi-Primary fuzzy Submodules"SC.Thesis. University of Baghdad. Mathematics | 207 2012( عام 1) ( العدد 30مجلة إبن الهيثم للعلوم الصرفة و التطبيقية المجلد ) Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.30 (1) 2017 انمودٌوالت انحذف شبه انتامة انضبابٍة حاتم ٌحٍى خهف جاهعت بغداد /للعلٌم الصزفت ) ابن الييثن (كليت الخزبيت /قسن الزياضياث هادي غانً راشذ ًسارة الخزبيت /األًلىحزبيت الزصافت 6102/ االول/كانون 62قبم فً: 6102/كانون األول/5استهم فً : انخالصة Q-FCF(M). ـشبابيت . ًقد حن إعطائيا الزهفي ىذا البحث حوج دراست فكزة الوٌديٌالث الحذف شبو الخاهت الض ًىي اعوام لحالت الوٌديٌالث الحذف شبو الخاهت االعخياديت ًقد حن حعوين ىذه الفكزه الى هٌديٌالث شبو الخاهت العظوى . نخائج عديدة ًخٌاص كثيزة حوج دراسخيا في بحثنا ىذا. Q-MFCF(M)الضبابيت ًقد حن اعطائيا الزهش بابيت , الوٌديٌالث الحذف شبو الخاهت االعخياديت , الوٌديٌالث الوٌديٌالث الحذف شبو الخاهت الض انكهمات انمفتاحٍة: الحذف شبو الخاهت لعظوى الضبابيت .