OSR Journal of Mathemats (OSR-JM) e-ssn: 2278-5728, p-ssn: 2319-765X. Volume 11, ssue 4 Ver. V (Jul - ug. 2015), PP 48-54 www.osrjournals.org Fuzzy Rngs and nt Fuzzy Rngs Wth Operators M.Z.lam Department of Mathemats,Mllat College, Darbhanga-846004 (ND) bstrat: n ths paper, we studed the theory of fuzzy rngs, the onept of fuzzy rng wth operators, fuzzy deal and ant-fuzzy deal wth operators, fuzzy homomorphsm wth operators et., and ther some elementary propertes. Keywords:Fuzzy rngs, fuzzy rng wth operators, fuzzy deal wth operators, ant-fuzzy deal, and homomorphsm.. ntroduton n 1982 W.J.Lu [1] ntrodued the onept of fuzzy rng. n 1985 Y.C.Ren [2] establshed the noton of fuzzy deal and quotent rng. n ths paper we studed the theory of fuzzy rng, and the onept of fuzzy rng wth operators, fuzzy deal and ant-fuzzy deal wth operators, homomorphsm and ther related elementary propertes have dsussed.. Prlmnares Defnton 2.1:[Lu [1]] Let R be a rng, a fuzzy set of R s alled a fuzzy rng of R f () (x y) mn ( (x), (y) ), for all x, y n R () (x y) mn ((x), (y) ), for all x, y n R Defnton 2.2: [Lu [1]] Let R be a rng, a fuzzy rng of R s alled a rng wth operator ( read as M fuzzy rng ) ff for any t [0,1], t s a rng wth operator of R (.e M subrng of R ), when t. Where t = {x R (x) t} Defnton 2.3: Let be M fuzzy deal of R, s a M fuzzy subrng of R suh that () (y + x y) (x) (v) (x y) (y) (v) ((x + z)y x y ) (z) For all x, y, z R Note that s a M fuzzy left deal of R f t satsfes (), (), () and (v), and s sad to be a M fuzzy rght deal of R, f t satsfes (), (), () and (v). Defnton 2.4: Let R be a rng, a fuzzy set of R s alled ant fuzzy subrng of R, f for all x, y, R (F 1 ) (x y) max { x, y } (F 2 ) (x y) max { x, y } Defnton 2.5: Let R be a rng, a fuzzy rng of R s alled an ant-fuzzy subrng wth operator ( read as ant M- fuzzy subrng) ff for any t [0,1], t s an ant-rng wth operator of R (.e.ant M-fuzzy subrng of R), when t. Where t = {x R (x) t } Defnton 2.6: Let be M-fuzzy ant deal of R, f s a ant M-fuzzy subrng of R suh that the followng ondtons are satsfed (F 3 ) (y + x y) (x) (F 4 ) (x y) (y) (F 5 ) ((x + z) y - x y) (z) For all x, y, z R. Note that s an ant M-fuzzy left deal of R f t satsfes (F 1 ), (F 2 ), (F 3 ) and (F 4 ), and s alled an ant M-fuzzy rght deal of R f t satsfes (F 1 ), (F 2 ), (F 3 ) and (F 5 ). DO: 10.9790/5728-11444854 www.osrjournals.org 48 Page
Fuzzy Rngs nd nt Fuzzy Rngs Wth Operators Example: Let R = { a 1, a 2, a 3, a 4 }, be a set wth two bnary operatons as follows + a 1 a 2 a 3 a 4 x a 1 a 2 a 3 a 4 a 1 a 1 a 2 a 3 a 4 a 1 a 1 a 1 a 1 a 1 a 2 a 2 a 1 a 4 a 3 a 2 a 1 a 1 a 1 a 1 a 3 a 3 a 4 a 2 a 1 a 3 a 1 a 1 a 1 a 1 a 4 a 4 a 3 a 1 a 2 a 4 a 1 a 1 a 2 a 2 Then (R,+,. ) s a rng. We defne a fuzzy subset : R 0,1, by (a 3 ) = (a 4 ) > (a 2 ) > (a 1 ). Then s an ant M-fuzzy rght (left) deal of rng R. Every ant M-fuzzy rght (left) deal of rng R s an ant M-fuzzy subrng of R, but onverse s not true as shown n the followng example. Example: Let R = { a 1, a 2, a 3, a 4 }, be a set wth two bnary operatons as follows + a 1 a 2 a 3 a 4 x a 1 a 2 a 3 a 4 a 1 a 1 a 2 a 3 a 4 a 1 a 1 a 1 a 1 a 1 a 2 a 2 a 1 a 4 a 3 a 2 a 1 a 2 a 3 a 4 a 3 a 3 a 4 a 2 a 1 a 3 a 1 a 1 a 1 a 1 a 4 a 4 a 3 a 1 a 2 a 4 a 1 a 1 a 1 a 1 Then (R, +,. ) s a rng. We defne a fuzzy subset : R 0,1, by (a 3 ) = a 4 > (a 2 ) > (a 1 ). Then s ant M-fuzzy subrng of R. But s not an ant M-fuzzy rght deal of R, sne ((a 1 + a 2 ) a 3 a 1 a 3 ) = a 3 > (a 2 ) Theorem 2.1: Let R be a M-rng and a fuzzy set on R s ant M-fuzzy rng n R f and only f s a M-fuzzy rng n R. Proof : Frst we suppose that be an ant M-fuzzy subrng n R. Then we have to show that s an M-fuzzy subrng n R. Letx, y R, we have x y 1 x y x y 1 max x, y mn 1 x,1 y mn x, y lso, xy 1 xy xy 1 max x, y mn 1 x,1 y mn x, y Therefore, s a M-fuzzy subrng n R. Conversely, suppose that s an ant M-fuzzy subrng n R, then we have to show that s a M-fuzzy subrng n R. Let x, y, R, we have x y 1 x y 1 max x y x, y mn 1 x,1 y mn x, y lso, xy 1 xy 1 max xy x, y DO: 10.9790/5728-11444854 www.osrjournals.org 49 Page
y Fuzzy Rngs nd nt Fuzzy Rngs Wth Operators mn 1 x,1 y mn x, Thus s M-fuzzy subrng n R. Theorem 2.2: Let R be a M- rng and be a M-fuzzy subrng n R.Then s an ant M-fuzzy deal n R f and only f s a M-fuzzy deal n R. Proof : Frst we suppose that s ant M-fuzzy deal n R, then we have to show that s a M-fuzzy deal n R. Let x, y, R, by defnton of ant M-fuzzy deal x y x x xr: x t, where t [0,1] Sne t 1 x y x 1 x x y x x lso, xy y 1 xy 1 y xy y nd x z y xy z 1 x z y xy 1 z x z y xy z Hene, satsfes all the ondton of M-fuzzy deal. Conversely, f we suppose that s an M-fuzzy deal n R, then we have to show that s ant M-fuzzy deal n R. Let x, y R and s an M-fuzzy deal n R, then by the defnton of M-fuzzy deal we have x y x x t : x y x x lso, xy y xy y nd x z y xy z x z y xy z xr x t, t [0,1] Hene, satsfes all the ondton of ant M-fuzzy deal n R. Defnton 2.7: Let R be a M-rng and a famly M-fuzzy deal of R defned by : Theorem 2.3: Let R be a M-rng, of R. Proof: Let x nf x :, for all x R : nf : : of M-fuzzy deal of R, then the nterseton be a famly of M-fuzzy deal of R, then s an M-fuzzy deal be a famly of M-fuzzy deal of R, and x, y R x y x y, for all x, y R nf mn x, y, then we have of DO: 10.9790/5728-11444854 www.osrjournals.org 50 Page
mn nf x : nf y : x y mn, lso, xy nf xy : x y nf mn x, y :, for all, Fuzzy Rngs nd nt Fuzzy Rngs Wth Operators R mn nf x :,nf y : x y mn, Hene s a M-fuzzy rng of R. For any x, y, z R we have nf x: y x y nf y x y : x xy nf xy : lso, nd Hene xy nf y : y x z y xy nf x z y xy : nf z: s a M-fuzzy deal of R. Defnton 2.8:Let R be a M-rng and of ant M-fuzzy deal of R defned by z : be a famly of ant M-fuzzy deal of R, then the unon : Theorem 2.4: Let R be a M-rng, and M-fuzzy deal of R. Proof :Let : x sup x :, for all x R be a famly of ant M-fuzzy deal of R, and x, y x y sup x y : supmax x, y : be a famly of ant M-fuzzy deal of R, then s an ant max sup x :,sup y : x y max, R,then we have DO: 10.9790/5728-11444854 www.osrjournals.org 51 Page
xy sup xy : lso Hene, x y sup max, : max sup x :,sup y : x y max,, for all x, y R s a ant M-fuzzy rng of R. For any x, y, z R, we have sup x: y x y sup y x y : nd x xy sup xy : Smlarly, Therefore, sup y : y sup z: x z y xy sup x z y xy : s an ant M-fuzzy deal of R z Fuzzy Rngs nd nt Fuzzy Rngs Wth Operators Defnton 2.9: Let R and S be two M-rng and f s a funton from R onto S () f B s a M-fuzzy rng n S, then the pre-mage of B under f s a M-fuzzy rng n R, defned by () 1 f B x B f x, for eah x R f s a M-fuzzy rng of R, then the mage of under f s a M-fuzzy rng n S defned by 1 sup x, f f y 1 f x f y y 0, otherwse For eah y S Theorem 2.5: Let R and S be M-rng and f : R S,be an M-homomorphsm from R onto S and () 1 f B s an M-fuzzy rng of S, then f B () f s an M-fuzzy rng of R, then f s an M-fuzzy rng of S. Proof: () Let x1, x2 R, then we have 1 2 1 2 mn B f x1, B f x2 1 1 mn f Bx, f Bx 1 f B x x B f x f x s an M-fuzzy rng of R. 1 2 DO: 10.9790/5728-11444854 www.osrjournals.org 52 Page
1 nd f Bx1. x2 B f x1. f x2 mn B f x1, B f x2 1 1 mn f Bx1, f Bx2 1 Hene, f B s a M-fuzzy rng of R. () Let y1, y2 S then we have Fuzzy Rngs nd nt Fuzzy Rngs Wth Operators 1 1 1 x : x f y1 y2 x1 x2 : x1 f y1, x2 f y2 1 Hene f y1 y2 sup x: x f y1 y2 1 1 sup x x : x f y, x f y 1 2 1 1 2 2 1 1 supmn x1, x2 : x1 f y1, x2 f y2 1 1 mnsup x1 : x1 f y1,sup x2 : x2 f y2 mn f y1, f y2 1 1 1 x : x f y1. y2 x1. x2 : x1 f y1, x2 f y2 1 1. 2 sup : 1. 2 1 1 sup x1. x2 : x1 f y1, x2 f y2 1 1 supmn x1, x2 : x1 f y1, x2 f y2 1 1 mnsup x1 : x1 f y1,sup x2 : x2 f y2 mn f y1, f y2 Sne Hene f f y y x x f y y s a M-fuzzy rng of S Theorem 2.6 :Let R and S be M-rng and f : R S be an M-homomorphsm from R onto S and. 1 f B s an M-fuzzy deal of S, then f B. f s an M-fuzzy deal of R, then f s an M-fuzzy deal of S 1 Proof: () Let B be an M-fuzzy deal of S, then f B s an M-fuzzy deal of R s an M-fuzzy rng of R from theorem 2.5() Let x1, x2, x3 R we have 1 f B x x x B f x f x f x 1 2 1 1 2 1 B f x 2 f 1 Bx 2 1 lso, 1 nd 1. 2 1. 2 B f x 2 f B x x B f x x f 1 Bx 2 f B x x x x x B f x f x f x f x f x 1 2 3 1 3 1 2 3 1 3 B f x 2 f 1 Bx 2 DO: 10.9790/5728-11444854 www.osrjournals.org 53 Page
Fuzzy Rngs nd nt Fuzzy Rngs Wth Operators 1 Hene, f B s an M-fuzzy deal of R. ()Let be an M-fuzzy deal of R, then f s an M-fuzzy rng of S from theorem 2.5().Now for any y1, y2, y3 S we have 1 f y1 y2 y1 sup x: x f y1 y2 y1 1 1 sup x1 x2 x1 : x1 f y1, x2 f y2 1 sup x2 : x2 f y2 f y 2 1 lso f y1. y2 sup x: x f y1. y2 1 1 sup x1. x2 : x1 f y1, x2 f y2 1 sup x2 : x2 f y2 f y 2 1 nd f y y y y y sup x : x f y y y y y 1 2 3 1 3 1 2 3 1 3 1 1 1 1 2 3 1 3 sup 1 2 3 1 3 : 1 1, 2 2, 3 3 f y y y y y x x x x x x f y x f y x f y 1 sup x2 : x2 f y2 f y 2 Hene f s an M-fuzzy deal of S. Referenes [1]. W.J.Lu, Fuzzy nvarant subgroups and fuzzy deals, Fuzzy sets and systems 8 (1982), 133-139. [2]. Y.C.Ren, Fuzzy deals and quotent rngs, Fuzzy Math 4 (1985), 19 26. [3]. T.K.Mukharjee and M.K.Sen, on fuzzy deals of rng, Fuzzy sets and systems 21 (1987), 99 104. [4]. W.X.Gu, S.Y.L and D.G.Chen, Fuzzy groups wth operators, Fuzzy sets and systems 66 (1994), 363 371. [5]. Q.Y.Xong, Modern algebra, (Sene and tehnology Publaton, Shangha,1978). DO: 10.9790/5728-11444854 www.osrjournals.org 54 Page