Microsoft Word - 115-119   115  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020       On intuitionistic Fuzzy Asly Ideal of Ring Showq Mohammed E Article history: Received 10 June 2019, Accepted 8 September 2019, Published in April 2020. Abstract In this paper we tend to describe the notions of intuitionistic fuzzy asly ideal of ring indicated by (I. F.ASLY) ideal and, we will explore some properties and connections about this concept. Key words: fuzzy set, intuitionistic fuzzy sub ring, intuitionistic fuzzy ideal. Introduction . 1 In 1965, Zadeh introduceed the notion of a fuzzy set [1]. In 1986 Rosenfeld applied this concept to group theory[2]. In 1986 Atanassov introduced the concept of intuitionistic fuzzy set . Let A in a non-empty set X is an object having the form A= }))(),(,{( Xxxxx AA  , where the functions ]1,0[: XA denote the degree of membership and ]1,0[: XA the degree of non-membership of each element Xx  to the set A and X xallfor 1)()(0  xAxA  [3]. In 1989 Biswas introduced the intuitionistic fuzzy subgroup and studied some of its properties [4]. In 2003 Banerjee and Basnet investigated intuitionistic fuzzy subrings and intuitionistic fuzzy ideal using intuitionistic fuzzy sets [5-7]. In this paper, we will recall some basic definitions. Let R be a ring, an (I.F.S) A=   )),( ),(,( RAA   of R is said to be intuitionistic fuzzy subring means by (IFS) of R if ].7,6.[, )},( ),( max{)( and )}( ),( max{)( )},(),(min{)( , )}(),(min{)( RAAA AAAAAAAAA       In 2012, sharma P.K introduced the notion of intuitionistic fuzzy ideal by (I.F.I). Let: A=  R)),( ),(,( AA   of a ring R if satisfies the four conditions, )},(),(max{)( , )}(),(min{)(  AAAAAA   )}(),(min{)( and )}(),(max{)(  AAAAAA   [8]. Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.2.2432 AL-Kufa university For Girls , ,College of Education Department of Mathematics showqm.ibriheem@uokufa.edu.iq   116 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 2. Intuitionistic Fuzzy Asly Ideal Definition (1) Let A=  )),( ),(,( RAA   be (IFS) of a ring R said to be an intuitionistic fuzzy asly ideal of a ring R means by (I. F.ASLY) ideal if and only if ].1,0[:)(],1,0[:)( ,)}.(),(min{)( 4. )}()( .3 )}(),(max{)( 2. )()( .1      RRwhere R AA AAA AA AAA AA           Example (2) Let R be the set of 22  matrices over non negative integer Z otherwise 9.0 }3,1/{, 0 0 q s if 2.0 )( otherwise 1.0 }3,1/{, 0 0 q s if 8.0 )(                           Zqswhere A Zqswhere A     Clearly   )),( ),(,( ZA AA   is an (I. F.ASLY ) ideal . Definition (3) Let A=     )),( ),(,(B , )),( ),(,( RBBRAA   are any two (I. F.ASLY ) ideals then their product is defined by: .,, )],( )([)( . )( )]( )([)(. )( . . RssBABA sBABA s s             Definition (4) Let A=     )),( ),(,(B , )),( ),(,( RR BBAA   are any (I. F.ASLY) ideals then their sum is defined by:  .),( )( ),( )(, RBA BABA   where .,,)],( )([)( )( )]( )([)(. )( Rss s BA s BA BA s BA             Theorem (5) Let A=   )),( ),(,( RAA   be (I. F.ASLY) ideal of a ring R and let  })),( ),(,( * A ** RA A   be the (IFS) of R is characterized by R, 1 )0()()( , )0(1)()( * *   AAAAAA then *A is (I. F.ASLY) an ideal of R. Proof For all R 1 )0()()( , )0(1)()( **  AAAAAA  we have   117 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 )1.....(..........).........( )0(-1 )( )0(-1 )( )( * *   AAA AAA    )2.........(..........)}........(),(max{ } )0(-1)(, )0(-1)({max )0(-1 )}(),({max )0(-1 )( )( ** *     AA AAAA AAA AAA         )3(..............................)}........() 1 )0(- )( 1)0(- )( )( * *   AAA AAA    )4( .........................)}........(),(min{ } 1)0(- )(, 1)0(- )(min{ 1 )0(- )}(),({min 1 )0(- )( )( ** *     AA AAAA AAA AAA         In forms (1),(2),(3) and (4), we have *A is (I. F.ASLY) ideal of R Theorem (6) Let A be (I. F.ASLY) ideal of R and let ]1,0[)]0(,0[: , ]1,0[)]0(,0[:  AA  be increasing functions, then (IFS)   )),( ),(,( RA AA    Means by )) ( ()( )) ,(( ) ( AAAA     is (I. F.ASLY) ideal of R. Proof R ,  .(1)....................).....( )( ))(( )(       AA AA   )..(....................) }.......(),({ ))}(()),(( { ))(()( AA AA AA 2max max            3).........(.................... )}()( ))(()(     AA AA    4).........(.................... )}(),({min ))}( ( )),( ({min ))( ()(      AA AA AA       In forms (1),(2),(3) and (4), we get fA an (I. F.ASLY) ideal of R . Theorem (7) If }{ jB j is a family of (I. F.ASLY) ideals of R, then  Jj jB is (I. F.ASLY) ideal of R.   118 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Proof Let }{ jB j be a family of (I. F.ASLY) ideals of R, for all R,  . We have  j j B =  )),( ),(,( R jBjB jj      )........(1....................)()( })({ )( inf inf            j BB j B jj B jj jj .(2)....................}.........)(),({max )}( ),( { max }))(),({(max )( supsup sup               j B j B B j B j BB jj B jj jj jjj    )........(3....................)( )( )( inf      j BB jj B jjj  .(4)....................}.........)(),({min )}( ),( { min }))(),({(min )( infinf inf               j B j B B j B j BB jj B jj jj jjj    In forms (1),(2),(3) and (4)  j j B =  ),(),(, R jB jj j B      is (I. F.ASLY) ideal of R. Theorem (8) Let A=  )),( ),(,( RAA   be a (I. F.ASLY) ideal of R, then 1. One of ).( ),(),(   AAA  at least two are equal. 2. One of ).( ),(),(   AAA  at least two are equal. Proof 1 If , )()( AA   so we have two cases: Case 1: if )()( AA   )}(),(max{).(  AAA   Then ).().(  AA   Case 2: if )()( AA   )}(),(max{).(  AAA   Then ).().(  AA   Proof 2 If )()( AA   Either )()( AA   ).().( )}(),(min{).(   AA AAA       119 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (2) 2020 Or )()( AA   ).().( )}(),(min{).(   AA AAA     References 1. Zadeh, L.A. Fuzzy Sets. Information and Control.1965, 8, 1, 338-353. 2. Rseronfeld R. Fuzzy group. J.Math. Anal.1971, 3, 5, 512-517. 3. Atanassov. Intuitionistic Fuzzy Sets. Fuzzy Sets and Systems.1986, 2010, 87-96. 4. Biswas, R. Intuitionistic Fuzzy Subrings. Mathematical Forum.1989, 10, 37-46. 5. Banerjee, B.; Basnet, D.K.R. Intuitionistic Fuzzy Subrings and ideals. J.Fuzzy Math. 2003, 11, 1,139-155. 6. Meena, K.; Thomas, V. Intuitionistic L-fuzzy Subrings. International mathematical frown.2011, 6, 53, 2561-2572. 7. Basnet, D.K. Topics in intuitionistic. fuzzy Algebra.2011, ISBN: 978-3-8443-9147-3. 8. Sharma, P.K. On Intuitionistic Anit. Fuzzy Ideal In Ring. Int. J. of Mathematical Sciences and Applications.2012, 2, 1, 139-146.