Fuzzy Relations on Modules

Transcription

1 International Journal of Algebra, Vol 3, 2009, no 20, Fuzzy Relations on Modules Bayram Ali Ersoy Yildiz Technical University, Department of Mathematics Davutpasa Kampusu, Esenler, Istanbul, Turkey Dan Ralescu University of Cincinnati, Cincinnati, OH , USA Abstract We investigate the fuzzy relations on modules similar Malik and Mordeson [4] and give some characterizations of fuzzy image and preimage of modules Mathematics Subject Classification: 03E72, 13A99, 13C05 Keywords: Fuzzy ideal, Fuzzy submodule, Fuzzy relation, Fuzzy image and Fuzzy preimage 1 Introduction As it is well known Zadeh[7], introduced the notion of a fuzzy subset of a nonempty set X as a function from X to unit real interval I = [0, 1] Fuzzy subgroup and its important properties were defined and established by Rosenfeld[1] Then many authors have studied about it After this time it was necessary to define fuzzy ideal of a ring The notion of a fuzzy ideal of a ring was introduced by Liu[12,13] Malik[3,4], Mordeson[3,4] and Mukherjee[11] have studied fuzzy ideals The concept of a fuzzy relation on a set was introduced by Zadeh[7] Bhattacharya and Mukherjee[10] have studied fuzzy relation on groups Malik and Mordeson [4] studied fuzzy relation on rings Fuzzy submodules of M over R were first introduced by Negoita and Ralescu [2] Pan [5] studied fuzzy finitely generated modules and fuzzy quotient modules (also see Sidky [6]) The aim of this paper is to investigate fuzzy relations on modules similar malik and mordesons [4] paper In this paper M, N are R-modules and R is a

2 1008 B A Ersoy and D Ralescu commutative ring A fuzzy relation on R is the fuzzy subset of R R Then we prove some theorems on fuzzy relations by using our definitionthat is; and are fuzzy submodules of M then the cartesian product of and ; is a fuzzy submodule of M M If is a fuzzy submodule of M M, then and are fuzzy submodules of M From these we conclude that, if and are fuzzy submodules of M then the level set of cartesian product of and ( ) t is a submodule of M M Moreover Let be a subset of M Then the strongest fuzzy subset is a fuzzy submodule of M M if and only if is a fuzzy submodule of M f : M M N N and be f invariant of M M and N N If is a fuzzy submodule of M M, then f ( ) is fuzzy submodules of N N Let f : M M N N and be f invariant of M M and N N If is a fuzzy submodule of N N, then f 1 ( ) is fuzzy submodules of M M Therefore, If f : M M N N is an homomorphism and is an f -invariant of M M and N N, then the mapping f ( ) defines a one-one correspondence between the set of all fuzzy submodules of M M and the the set of all fuzzy submodules of N N 2 Preliminaries We will give some basic definitions and theorems Definition21[7]: A fuzzy subset of R is a function : R [ 0,1] Definition22[13]: A fuzzy subset of R is called a fuzzy left (right) ideal of R if (i) ( x y) min( ( x), ( y)) (ii) ) ( xy) ( y) ( ( xy) ( x)) for all x, y R A fuzzy subset of R is called a fuzzy ideal of R if is a fuzzy left and fuzzy right ideal of R Definition23[13]: If is a fuzzy subset of R, then for any t Im, the set x R ( x t t = ) is called the level subset of R with respect to Theorem24[13]: Let be fuzzy subset of R is a fuzzy ideal of R if and only if t is an ideal of R for t Im Definition25[4]: A fuzzy relation on M is a fuzzy subset of M M Definition26[4]: Let be a fuzzy relation on M and let be a fuzzy subset of M Then is called a fuzzy relation on if x, y M ( x, y) min( ( x), ( y)) Definition27[4]: Let and be a fuzzy subset of M The cartesian product of and is defined by, x, y M ( x, y) = min( ( x), ( y))

3 Fuzzy relations on modules 1009 Definition28[4]: Let be a fuzzy subset of M Then the strongest fuzzy subset on M that is a fuzzy relation on is defined by, x, y M ( x, y) = ( x, y) = min( ( x), ( y)) Definition29[4]: Let be a fuzzy relation on M Then the weakest fuzzy subset of M on which is a fuzzy relation is defined by, x M ( x) = sup max ( xy, ), ( yx, ) ( x, y) = min( ( x), ( y)) y M Definition210[4]: Let and be a fuzzy subset of M Then, x M ο ( x, y) = sup min( ( y), ( z)) Definition211[3]: Let f : M N and be fuzzy subset of M The fuzzy subset f ( ) of N defined as follows; for all y N, x= yz 1 ( x): x M, f( x) = y if f ( y) f( )( y) = is called the fuzzy 0 otherwise image of under f Definition212[3]: Let f : M N and be fuzzy subset of N The fuzzy subset 1 f ( ) of N defined as follows; for all x M, f 1 ( )( x) = ( f( x)) is called fuzzy preimage of under f Definition213[3]: Let M and N be two R-modules The direct sum of M and N, denoted M N, is the R-module, which as a set is the Cartesian product of M and N, with addition and multiplication defined coordinate by coordinate: (m 1, n 1)+(m 2, n 2) = (m 1+m 2,n 1+ n 2) and r(m, n) = (rm, rn) Definition214[5,6]: A fuzzy submodule of M is a fuzzy subset of M such that 1- (0) = 1 2- ( rx) ( x) r R and x M 3- ( x + y) min( ( x), ( y)) x, y M Theorem215[5,6]: is a fuzzy submodule of M if and only if t Im t is a submodule of M 3 Fuzzy relations on modules Malik and Mordeson [ ] analised fuzzy relation on fuzzy rings and groups Similarly we get new useful result in fuzzy module theory Theorem31: and are fuzzy submodules of M then the cartesian product of and is a fuzzy submodule of M M Proof: Since ( o) = 1 = (0) 1- (0,0) = min( (0), (0)) = 1

4 1010 B A Ersoy and D Ralescu ( rx, ry) = min( ( rx), ( ry)) 2- min( ( x), ( y)) ( xy, ) (( xy, ),( z+ t)) = ( x+ zy, + t) = min( ( x+ z), ( y+ t)) 3- min( ( x), ( y))min, ( y) ( t)) = min( ( x), ( y))min( ( z), ( t)) = min( ( xy, ), ( zt, )) Therefore is a fuzzy submodule of M M Theorem32: If is a fuzzy submodule of M M then and are fuzzy submodules of M Proof: (0,0) = min( (0), (0)) = (0) = 1 and (0) = 1 ( rx) ( rx,0) (( r,0)( x,0)) ( x,0) = min( ( x), (0)) = ( x) ( x+ y) ( x+ y,0) (( x,0) + ( y,0)) min( ( x,0), ( y,0)) = min( ( x), ( y)) Then is fuzzy submodules of M One can show by the same way that is fuzzy submodule of M Theorem33: Let and are fuzzy submodules of M then the cartesian product of and is a fuzzy submodule of M M if and only if ( ) t is a submodule of M M for all t [ 0,1] is a fuzzy submodule of M M Proof: Trivial Lemma34: If is fuzzy submodule of M and = then = Proof: ( x) ( x,0) = min( ( x), (0)) = min( ( x,0), (0,0)) = min( ( x), (0)) = ( x)

5 Fuzzy relations on modules 1011 Corollary35: Let be a subset of M and only if is a fuzzy submodule of M is a fuzzy submodule of M M if We now give the one-one correspondence between all fuzzy submodules M M and N N Theorem36: Let f : M M N N and be f invariant of M M and N N If is a fuzzy submodule of M M then f ( ) is fuzzy submodules of N N Proof: f ( )(0,0) = sup ( (0,0) f( x, y) = (0,0) ( x, y) M M = sup min( (0), (0)) f( x, y) = (0,0) = sup 1,1 = 1 For r1, r2r 1r2 R, x1, x2 N and x1x2 M f( )( rx, r x ) = sup ( r x, r x ) f( r x, r x ) = ( rx, r x ) rx rx 2 2 f rx rx 2 2 rx rx 2 2 ` ` x1 x2 x1 x2 f x1 x2 x1 x2 = sup min( ( ), ( )) (, ) = (, ) sup min( ( ), ( )) = sup min( (, ) (, ) = (, ) = f( )( x1, x2) For x1, y1, x2, y2 N x1, y1, x2, y2 M f( )(( x1, x2) + ( y1, y2)) = f( )( x1 + y1, x2 + y2) = sup ( x + y, x + y ) f( x + y, x + y ) = ( x + y, x + y ) x1 y1 x2 y2 x1 y1 x2 y2 x1 x2 y1 y2 x1 x2 y1 y2 f x1 x2 x1 x2 f y1 y2 y1 y2 = sup min( ( + ), ( + )) sup min( ( ), ( )), min( ( ), ( )) = sup min( ( ), ( )), min( ( ), ( )) = sup ((, )), ((, )) (, ) = (, ), (, ) = (, ) = min( f( )( x1, x2), f( )( y1, y2) Therefore f ( ) is fuzzy submodules of N N Theorem37: Let f : M M N N and be f invariant of M M and N N If is a fuzzy submodule of N N then submodules of M M Proof: f 1 ( ) is fuzzy

6 1012 B A Ersoy and D Ralescu f 1 ( )(0,0) ( f(0,0)) (0,0) = min( (0), (0)) = 1 r, r r r R, x, x M and x x N f rx r2x2 f rx r2x2 For ( )(, ) ( (, )) ( rx, rx ) 1 = f = min( ( rx), ( rx ) ) min( ( x ), ( x ) ) ` ` 1 2 ( x, x ) ` ` 1 2 ( f( x, x )) x, y, x, y M x, y, x, y N 1 2 ( )( x, x ) 1 2 f ( )(( x, x ) + ( y, y )) = f ( )(( x + y ),( x + y )) = ( )( f(( x + y ),( x + y )) (( x + y ),( x + y )) = min( ( x + y ), ( x + y ) ) min( ( x ), ( y )), min( ( x ), ( y )) = min( ( x ), ( x )),min( ( y ), ( y )) (( x1, x2 )), (( y1, y2 )), = ( f( x, x )), ( f( y, y )) = f ( )( x, x ), f ( )( y, y ) Hence f 1 ( ) is fuzzy submodules of M M Corollary38: f : M M N N is an homomorphism and is an f - invariant of M M and N N The mapping f ( ) defines a one-one correspondence between the set of all fuzzy submodules of M M and the the set of all fuzzy submodules of N N Proof: The immediate result of above theorems 4 Conclusion We get the fuzzy relation on fuzzy submodules That is; If and are fuzzy submodules of M if and only if the cartesian product of and is a fuzzy submodule of M M We investigate the strongest and the weakest fuzzy

7 Fuzzy relations on modules 1013 submodule Finally we have the one one correspondence between fuzzy submodules of fuzzy images and fuzzy preimages Acknowledgement Bayram Ali Ersoy's work was supported by the Scientific and Technological Research Council of Turkey(TUBITAK); Dan Ralescu's work was supported by a Taft Travel for Research Grant " References [1]A Rosenfeld, Fuzzy groups, J Math Anal Appl 35 (1971) [2]CV Negoita, and DA Ralescu, Application of fuzzy systems analysis, Birkhauser, Basel, 1975 [3] DS Malik, and J N Mordeson, Fuzzy Commutative Algebra, World scientific Publishing, 1998 [4]DS Malik, and J N Mordeson, Fuzzy relations on rings and groups, Fuzzy sets and systems 43 (1991) [5] FZ Pan, Fuzzy finitely generated modules, Fuzzy Sets and Systems, 21 (1987) [6] FI Sidky, On radical of fuzzy submodules and primary fuzzy submodules, Fuzzy Sets and Systems 119 (2001) [7] L A Zadeh, Fuzzy sets, Inform Control 8 (1965) [8] R, Kumar, Certain fuzzy ideals of rings redefined, Fuzzy sets and systems 46 (1992) [9]R Kumar, VN Dixit, and N Ajmal, On fuzzy rings, Fuzzy sets and systems 49 (1992) [10] TK Mukherjee, and P Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Inform Sci 34, (1984) [11] TK Mukherjee, and MK Sen, On fuzzy ideals of a ring I, Fuzzy sets and systems 21 (1987) [12] W Liu, Fuzzy Invariant subgroups and fuzzy ideals, Fuzzy sets and systems 8, (1982) [13]W Liu, Operations on fuzzy ideals, Fuzzy sets and systems 11, (1983) 31-41

8 1014 B A Ersoy and D Ralescu [14] Y Alkhamees, and J N Mordeson, Fuzzy Localized Subrings, Information Sciences 99, (1997) [15] Y Alkhamees, and J N Mordeson, Fuzzy principal ideals and fuzzy simple field extensions, Fuzzy sets and systems 96, (1998) Received: March, 2009