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J.Thi-Qar Sci. Vol.1 (1) March/2008 ISSN 1991- 8690 1661 - 0968 الترقيم الدولي On Quasi Prime Fuzzy Submodules and Quasi Primary Fuzzy Submodules Rabee Hadi Jari Hassan Kiream Hassan Department of mathematics - College of Education, University of Thi-Qar thiqaruni.org. Abstract In this paper we give some results about fuzzy quasi prime submodules, also,we study the notion of quasi primary fuzzy 0 State in 18O by using Core – Polarization submodules. 2 effects Emad A. Salman Department of Physics, College of Science, Thi-qar University 1. Introduction: We record here some basic The notion of prime fuzzy concepts and clarify notions which are submodules was introduced by Hatem used in the next work .For details,we Y.K [1] and some results a bout this refer to [2,3]. concept are presented .In this paper Definition 2.1 [2]: we give more results about this A fuzzy set is a mapping X from a concept and study the extension of nonempty set M in to [0,1], then the quasi primary fuzzy submodule by set fuzzifing the notion of (ordinary) quasi primary fuzzy submodule and give some characterizations about it is called a level subset of X. ,also we give some properties related Definition 2.2 [3]: to it. Let be a fuzzy set Through out the paper R is a commutative ring with unity and M is in M, where, x M , t [0,1] , then xt is an R-module. 2. Preliminaries: J.Thi-Qar Sci. Vol.1 (1) March/2008 defined by : Where A is a proper fuzzy submodule t if x y of X. xt ( y ) o if x y for all y M. x t is called a fuzzy singleton. 3. Quasi Prime Fuzzy submodules: If x=0 and t=1, then: In this section we turn our attention to the extension of a quasi Definition 2.3 [4]: prime modules. A fuzzy set X in R-module M is Definition 3.1 [1]: called fuzzy R-module if and only if A fuzzy submodule A of a fuzzy each x, y M and r R , then: module X of an R-module M is called 1 if y 0 quasi prime fuzzy submodule of X if 01 ( y) whenever a s bk xt A for fuzzy 0 if y 0 Definition 2.3 [4]: singleton a s , bk of R and x t X , A fuzzy set X in R-module M is implies that a s bk A or b k xt A . called a fuzzy R-module if and only if Definition 3.2 [1]: each x, y M and r R , then: Let X be a fuzzy module of an R- 1. X ( x y) min X ( x), X ( y). module M. X is called a quasi prime 2. X (rx) X ( x) fuzzy module if and only if F-annA is 3. X(0)=1. (0 is the zero element of M). a prime fuzzy ideal of R for each Definition 2.4 [5]: nonempty fuzzy submodule A of X. Let A and X be two fuzzy Equivalently, F-ann(xt)=F-ann(rsxt) module of an R-module M, A is called for each xt X and for each fuzzy a fuzzy submodule of X if and only if singleton rs of R such that A X . r s F annX . Definition 2.5 [5]: Definition 3.3 [6]: Let rt and xs be two fuzzy Let A be a fuzzy submodule of a singletons of R and M respectively. fuzzy module X, then A is called an then rt x s (rx ) , Where =min{s,t}. essential fuzzy submodule if Definition 2.6 [5]: A B 01 ,for each nonempty fuzzy Let rt be a fuzzy singleton of R submodule B of X. and A be a fuzzy module of R-module. Lemma 3.4[6]: Then for any Let A be an essential fuzzy w M , submodule of X,if a k X ,then there sup{inf{t,A(x)}},if all w rx for some x M. (rt A)(w) exists a fuzzy singleton rt of R such o otherwise that 01 rt a k A. we noticed that if a fuzzy module X Definition 2.7 [6]: is quasi prime fuzzy module ,then any Let X be a nonempty fuzzy fuzzy submodule of X is a quasi prime module of R-module M. The fuzzy fuzzy module but the converse is not annihilator of A denoted by F-annA is true,see[1,Remarks and Examples defined by: 2.1.3(4)]. However, we put certain F annA (01 : A) xt : xt A 01 , x R, t [0,1] condition under which the converse is true. J.Thi-Qar Sci. Vol.1 (1) March/2008 Proposition 3.5: R if for each rt a k I ,then either Let A be an essential fuzzy rt I or a k I ,[7]. submodule of a fuzzy module X such that F-ann xt=F-ann rsxt for each Corollary 3.7: Let A be an essential fuzzy xt X and for a fuzzy singleton rs of submodule of a fuzzy module X, if A is R such that rs F annxt , then X is a a faithful quasi prime module, then X quasi prime fuzzy module if and only is a faithful quasi prime fuzzy module. if A is a quasi prime fuzzy module. Proof: Proof Since A is a faithful, then F- Suppose that A is a quasi prime annA=01, also, A X , then fuzzy R-module. F annX F annA by [6, Remark To prove that 01 is a quasi prime fuzzy 1.2.18], which implies that submodule of X. F annX 01 , but 01 F annX , hence F-annX=01, That is X is faithful, Let al bk xt 01 moreover it follows that F-annX=F- for fuzzy singletons al, bk of R and annA. xt X .Since A is an essential fuzzy Now, to prove X is a quasi prime fuzzy submodule, thus by lemma 3.4, there module,it is enough to prove that F- exists a fuzzy singleton rs of R such ann(xt)=F-ann(rsxt) for each xt X that 01 rs xt A ,that is and for each fuzzy singleton rs of R rs F annxt . such that r s F annX . Hence al bk (rs xt ) 01 . But A is a quasi Let al F annrs xt for a fuzzy prime fuzzy R-module, thus 01 is a singleton rs of R and xt X , then quasi prime fuzzy submodule of A by al (rs xt ) 01 . [1, proposition 3.4.1], so either al (rs xt ) 01 or bk (rs xt ) 01 , thus Consequently, (al rs ) xt 01 . This either al F annrs xt or implies that al rs F annxt , so bk F annrs xt . But al rs F annX . But F-annX=F- F annrs xt F annxt . annA,then al rs F annA , thus F- Consequently, either al F annxt or annA is prime fuzzy ideal by [1, bk F annxt , that is either al xt 01 definition 2.1.1], since rs F annA. or bk xt 01 . Hence X is a quasi prime So a l F annA , implies that fuzzy module by [1, proposition 3.4.1]. al F annX , hence al X 01 , that The converse follows directly by is al xt 01 for all xt X , hence [1, Remarks and Examples 2.1.3(4)]. Definition 3.6 [6]: F annrs xt F annxt …(1) A fuzzy module X is called faithful Now, let al F annxt for a fuzzy if F-annX=01. singleton al of R and xt X . Recall that a fuzzy ideal I of a By lemma 3.4, there exists fuzzy ring R is called a prime fuzzy ideal of singleton rs of R such that 01 rs xt A , that is rs F annxt , J.Thi-Qar Sci. Vol.1 (1) March/2008 so al rs xt 01 which implies that In this section we introduce the al F annrs xt , thus concept of quasi primary fuzzy submodule by fuzzifying the ordinary F annxt F annrs xt …(2) notion of primary submodule . From (1) and (2) we get F-annxt=F- Definition 4.1 [5]: annrsxt, thus X is a quasi prime fuzzy A fuzzy submodule A of a fuzzy module by [1, theorem 2.1.9(3)]. module X is called a primary fuzzy Definition 3.8 [6]: submodule if for each fuzzy singleton A fuzzy module X is called fuzzy rt of R and a k X such that torsion free if F-annxt=01 for any rt a k A , then either x t X , x t 01 where F annxt rk : rk fuzzy singleton of R, rk xt 01 . Lemma 3.9: ,where Let R be an integral domain and ( A : X ) {rt : rt X A, rt fuzzy sin gletonof R} let X be a torsion free fuzzy module of Recall that if A is a fuzzy R-module M, then X is a quasi prime submodule of a fuzzy module X of an fuzzy module. R-module and I is a fuzzy ideal of R Proof: then (A:I) define as: Suppose that X is a torsion free ( A : I ) {a k , a k I A, a k X } ,note fuzzy module. To prove 01 is quasi that :If I=(rt),then: prime fuzzy submodule of X. ( A : (rt )) {a k : rt a k A, a k X } ,[8]. Let al bk xt 01 ,for fuzzy singletons Proposition 4.2: al,bk of R and xt X . Let A be a proper fuzzy If submodule of a fuzzy module X, then bk xt 01 , then the following statements are al F annbk xt , since equivalent: 1. A is a primary fuzzy submodule of F ann(bk xt ) F anny , where X. min k, t and y=bx by [1, Remark 2. (A:I) is a primary fuzzy submodule 1.2.14] so al F anny . But of X for each fuzzy ideal I of R. 3. (A:(rt)) is a primary fuzzy F anny 01 because X is a torsion submodule of X for each fuzzy free, so a l 01 . singleton rt of R. On the other hand, 01 F annX , Proof: which implies that al F annX , 1 2: Suppose that rt x k ( A : I ) for each that is x k X and a fuzzy singleton rt of R al X 01 , by [6, definition 1.2.17], and I be a fuzzy ideal of R.It follows hence al xt 01 for all xt X . that al rt x k A for each fuzzy Thus, X is a quasi prime fuzzy module singleton al of I, then either al x k A by [1,proposition 3.4.1]. for each fuzzy singleton al of I or rt n ( A : X ) , n Z . 4. Quasi Primary fuzzy submodule. J.Thi-Qar Sci. Vol.1 (1) March/2008 So, either x k ( A : I ) or rtn X A, Now, let ak A : rt B for all since A ( A : I ), by [8, theorem 3.3], rt B A , hence a k ( A : rt B ) n for hence rt n X ( A : I ) , implies that some n Z and so a kn rt B A ,which implies that rt n (( A : I ) : X ) ,it follows, either a k rt ( A : B ) . But (A:B) is a primary n x k ( A : I ) or r (( A : I ) : X ) , hence n t fuzzy ideal of R and rt ( A : B) ,so a (A:I) is a primary fuzzy submodule of n m X. k ( A : B) for some m Z ,thus 2 3: a ( A : B) s k where s=nm so that It is clear. 3 1: ak A : B ; that is It follows easily by taking r=1 and t=1. A : rt B A : B and hence Recall that a proper submodule N A : B A : rt B . ofan R-module M is said to be quasi primary submodule if [N:K] is a primary ideal of R if for each Conversely, to prove (A:B) is a submodule K of M such that primary uzzy ideal of R for each N K ,where A B. [N:K]={r R:rk N},[9]. Suppose that a k rt ( A : B) for a fuzzy We shall fuzzify this concept as singleton ak and rt of R. If rt ( A : B) , follows: then ak ( A : rt B) A : rt B = A : B Definition 4.3 : A proper fuzzy submodule A of a and hence A is a quasi primary fuzzy fuzzy module X is said to be quasi submodule of X. primary fuzzy submodule if (A:B) is a Proposition 4.5: primary fuzzy ideal of R for each Let A be a fuzzy submodule of a fuzzy submodule B of X such that fuzzy module X, then A is a quasi A B where primary submodule of X if and only if (A:I) is a quasi primary fuzzy ( A : B) rt / rt B A, for a fuzzy singleton rt of R submodule of X for each fuzzy ideal I of R. . Proof: Proposition 4.4: Suppose that A is a quasi primary Let A be a proper fuzzy fuzzy submodule of X. submodule of a fuzzy module X, then By [8, theorem 3.3], (A:I) is a fuzzy A is a quasi primary fuzzy submodule submodule of X. of X if and only if A : B A : rt B To prove (A:I) is a quasi primary for each primary fuzzy submodule B fuzzy submodule of X, we must prove of X and for each fuzzy singleton rt of ((A:I):B) is a primary fuzzy ideal for R such that A B and rt B A . each ( A : I ) B. Proof: A ( A : I ) B by [8, lemma 2.16(1)] . It is clear that A : B A : rt B . Now, suppose that a k rt (( A : I ) : B) for fuzzy singleton rt and ak of R, then J.Thi-Qar Sci. Vol.1 (1) March/2008 a k rt B ( A : I ), so a k rt BI A , suppose Conversely, if 01 is quasi primary that a k (( A : I ) : B) , then a k BI A , fuzzy submodule of X, to prove X is a quasi primary fuzzy module. Since 01 hence a k rt I ( A : B ) . is a quasi primary fuzzy submodule, But, (A:B) is a primary fuzzy ideal by then (01:A) is a primary fuzzy ideal for definition 4.3 and a k I ( A : B ) .Thus, each nonempty fuzzy submodule. rt n ( A : B ) for some n Z and so But (01:A)=F-annA, so X is a quasi primary fuzzy R-module. rt n B A ( A : I ) , therefore rt (( A : I ) : B ) and hence (A:I) is a n REFRENCES: quasi primary fuzzy submodule of X. [1] Hatem Y.K.,(2001), "Fuzzy Quasi –prime The converse follows by taking Modules and Fuzzy Quasi –prime I R , where R (a) 1 for all a R submodules ", M.Sc. thesis, and by similar to the proof of [6, University of Baghdad, college of Education Ibn- AL- Haithem ,(69)p. proposition 2.1.16]. [2] Martines L.,(1996), "Fuzzy modules over Recall that an R-module m is said fuzzy rings in connection with Fuzzy to be a quasi primary module if annN ideal of Fuzzy ring ". J.Fuzzy is a primary ideal of R, for each non Mathematics, Vo1.4:843-857. zero submodule N of M,[9]. [3] Liu W. J,(1982),”Fuzzy invariant Our attempt to fuzzify this concept subgroups and Fuzzy ideals”, fuzzy is as follows: sets and systems,Vol.8:133-139. Definition 4.6 : [4] Nanda S. (1989),"Fuzzy Modules over A fuzzy module X of R –module Fuzzy rings ", Bull. coll. Math. Soc., Vo1. 18 :197-200. M is said to be quasi primary fuzzy [5] Mashinchi M. and Zahedi M.M.,(1992)" module if and only if F-annA=(01:A) is On L-Fuzzy primary submodules ", a primary fuzzy ideal for each Fuzzy sets and systems, Vol.49:231- nonempty fuzzy submodule A of X . 236. [6] Jari R.H.(2001)," prime Fuzzy submodules and prime Fuzzy Proposition 4.7: modules ", M.Sc. Thesis, A fuzzy module X is a quasi University of Baghdad, primary fuzzy module if and only if 01 college of Education, Ibn –AL- is a quasi primary fuzzy submodule of Haithem,(64)p. X. [7] Mukhrjie T.K.,(1989)” Prime Fuzzy Proof: Ideals in Rings”, Fuzzy Sets and Suppose that X is a quasi primary Systems Vol. 32:337-341. fuzzy module. [8] Zahedi M.M.,(1992),"On L-Fuzzy To prove 01 is a quasi primary fuzzy Residual Quotient submodule of X, since X is a quasi Modules", Fuzzy sets and systems, Vol.51:333-344. primary fuzzy submodule, then F- [9] Abdul-al-Kalik A.J.,(2005),”Primary annA is a primary fuzzy ideal for each Modules”, M.Sc. thesis, University of nonempty fuzzy submodule A of Baghdad,college of Education X.Thus (01:A) is primary fuzzy ideal Ibn- AL-Haithem,(77)p. and hence 01 is quasi primary fuzzy submodule of X. .J.Thi-Qar Sci )1( 1.Vol 8002/March On Quasi Prime Fuzzy Submodules and Quasi Primary Fuzzy Submodules حسن كريم حسن ربيع هادي جاري قسم الرياضيات - كلية التربية - جامعة ذي قار الخالصة: في هذا البحث اعطينا بعض النتائج عن المقاسات الجزئية شبة االولية الضبابية ,كذلك درسنا مفهوم المقاسات الجزئية شبه االبتدائية الضبابية .

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