S. Ostadhadi-Dehkordi and B. Davvaz* Department of Mathematics, Yazd University, Yazd, Iran

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1 IJST (03) 37A3: 5-63 Iri Jourl of Siee & Tehology Idel theory i -semihyperrigs S Ostdhdi-Dehkordi d B Dvvz* Deprtmet of Mthemtis, Yzd Uiversity, Yzd, Ir E-mil: dvvz@yzdir Astrt The oept of -semihyperrig is geerliztio of semirig, geerliztio of semihyperrig d geerliztio of -semirig Sie the theory of idels plys importt role i the theory of - semihyperrig, i this pper, we will mke itesive study of the otios of Noetheri, Artii, simple d regulr -semihyperrigs The ulk of this pper is devoted to sttig d provig logues to severl theorems i the theory of -semihyperrigs Keywords: Semihyperrig; Γ-semihyperrig; Γ-hyperrig; simple Γ-semihyperrig; regulr Γ-hyperrig Prelimiries d si defiitio I 964, Nousw [] itrodued -rigs s geerliztio of terry rigs Bres [] slightly wekeed the oditios i the defiitio of -rig i the sese of Nousw Bres [], Luh [3] d Kyuo [4] studied the struture of -rigs d otied vrious geerliztios logous to orrespodig prts i rig theory The hyperstruture theory ws or i 934, whe the otio of hypergroup ws itrodued [5] Oe of the first ooks, dedited espeilly to hypergroups, is "Prolegome of Hypergroup Theory" writte y Corsii i 993 [6] Aother ook o "Hyperstrutures d Their Represettios", y Vougiouklis, ws pulished oe yer lter [7] We metio here other importt ook for the pplitios i Geometry d for the leress of the expositio, writte y W Preowitz d J Jtosik [8] Aother ook [9] is devoted espeilly to the study of hyperrig theory Severl kids of hyperrigs re itrodued d lyzed The volume eds with outlie of pplitios i hemistry d physis, lyzig severl speil kids of hyperstrutures: e - hyperstrutures d trspositio hypergroups The theory of suitle modified hyperstrutures serve s mthemtil kgroud i the field of qutum ommuitio systems Algeri hyperstrutures re suitle geerliztio of lssil lgeri strutures I lssil lgeri struture, the ompositio of two elemets is elemet, while i lgeri hyperstruture, the ompositio of two elemets is *Correspodig uthor Reeived: 9 Novemer 0 / Aepted: 6 Deemer 0 set More extly, let H e o-empty set The, the mp : H H P * ( H) is lled hyperopertio whe P * ( H ) is the fmily of oempty susets of H Let H e o-empty set d : H H P ( H) e hyperopertio The ouple ( H, ) is lled hypergroupoid ( H, ) is lled semihypergroup if for every x, y, z H, we hve x ( y z ) = ( x y) z Moreover, if for every x H, x H = H = H x, the ( H, ) is lled hypergroup Also, my uthors studied differet spets of semihypergroups, for iste, Bosig d Corsii [0,, ], Dvvz [3], Dvvz d Pourslvti [4], Fsio d Frei [5], Gut [6] d Leoreu [7] We sy tht hypergroup ( H, ) is oil if () it is ommuttive ( x y = y x, for every x, y H ), () it hs slr idetity (lso lled slr uit), whih mes tht e H x H, e x = x e = x, (3) every elemet hs uique iverse, whih mes tht for ll x H, there exists uique x H, suh tht e x x, (4) it is reversile, whih mes tht if x y z, y z the there exist the iverses of y d of z, suh tht z y x d y x z The otio of multiplitive hyperrig ws itrodued y Rot [8] i 98 The multiplitio is hyperopertio, while the dditio is opertio, tht is why it ws lled multiplitive hyperrig A triple ( R,, ) is lled

2 IJST (03) 37A3: multiplitive hyperrig if () ( R, ) is eli group, () ( R, ) is semihypergroup, 3) for ll,, R, we hve ( ) d ( ) If i (3) we hve equlities isted of ilusios, the we sy tht the multiplitive hyperrig is strogly distriutive The otio of -semirig ws itrodued y Ro [8, 9] s geerliztio of -rig s well s of semirig For exmple, let S e the dditive ommuttive semigroup of ll m mtries over the set of ll o-egtive itegers d e the dditive ommuttive semigroup of ll m mtries over the sme set The, S is - semirig if deotes the usul mtrix produt of,, where, Sd Dutt d Srdr [0] gve the meig of left d right opertor semirigs for give -semirig Let ( R,, ) e ritrry semirig d ={} It is esy to see tht R is -semirig Thus semirig e osidered s -semirig My lssil otios of semirig hve ee exteded to - semirig I [, ], Dvvz et l studied the otio of -semihypergroup s geerliztio of semihypergroup My lssil otios of semigroups d semihypergroups hve ee exteded to -semihypergroups d lot of results o -semihypergroups re otied Let ( R, ) e hypergroupoid d e oempty set The, R is lled - hyperrig if there exists mppig RR R (imges deoted y for ll, R, d ) stisfyig the followig oditios: () ( R, ) is oil hypergroup, () there exists zero elemet tht ilterlly sorig elemet, ie, x 0=0 x =0, x 0= x, for every d x R, () ( ) =, () ( ) =, (3) ( ) = ( ) Let R e ommuttive semihypergroup d e ommuttive group The, R is lled semihyperrig if there exists mp RR P * ( R) (imge to e deoted y for, Rd ) d P * ( R) is the set of ll o-empty susets of R stisfyig the oditios 3, 4, 5 d ( ) = for every, Rd, Let R e -semihyperrig The, ( R, ) is semihypergroup for every (Let e fixed elemet i We defie = for ll, S) I the ove defiitio, if R is semigroup, the R is lled multiplitive - semihyperr ig A -semihyperrig R is lled ommuttive if x y = y x for every x, y Rd We sy tht -semihyperrig R with zero, if there exists 0 R suh tht 0 d 00, 0 0 for ll R d Let A d B e two oempty susets of -semihyperrig R We defie A B = tr t A, B, AB ={ tr t A, B, }, A B = tr t, A, B,,, i= i i i i i i i= i i i i X = tr t x x X,, A o-empty suset R of -semihyperrig R is lled su-semihyperrig if it is losed with respet to the multiplitio d dditio I other words, o-empty suset R of -semihyperrig R is su -semihyperrig if R R R d R A right (left) idel I of - semihyperrig R is dditive su semihypergroup ( R, ) suh tht IR R ( RI I ) If I is oth right d left idel of R, the we sy tht I is two-sided idel or simply idel of R Let X e o-empty suset of - semihyperrig R By the term left idel < X > l (respetively, right idel < X > r ) of R geerted y X, tht is, the itersetio of ll left idels (respetively, right idels) of R otis X Hee, () < X > l = X R X, () < X > r = X X R, (3) < >= X R X X R X R A o-empty suset I of -hyperrig R is left (right) idel if d oly if (), I implies I, () I, r R d imply r I( r I) Let I e idel of -hyperrig R suh tht x I x I for ll x R The, I is lled orml idel of R If I is orml idel of - hyperrig R, the we defie the reltio x ymodi ( ) if d oly if ( x y ) I This reltio is deoted y xi y We defie the followig opertio d hyperopertio o the set of ll lsses [ R : I ]={ I ( x) x R}, s follows: I ( x) I ( y)={ I ( z) zi ( x) I ( y)}, I ( x) I ( y)= I ( xy) The, [ R: I ] is -hyperrig of whih

3 53 IJST (03) 37A3: 5-63 ={ } A idel I of -hyperrig R is lled ull if I I ={0} d is lled idempotet if I I = I Let ( R, ) d ( R, ) e two - d - semihyperrigs, respetively d f : e mp The, : R Ris lled (, ) - homomorphism or (shortly, homomorphism), if for every x, y Rd, () ( x y) ={ ( t) t x y} ( x) ( y), () ( x y)={ ( t) t xy} ( x) f ( ) ( y), (3) f ( x y)= f ( x) f ( y) I the ove defiitio, if ( x y)= ( x) ( y) d ( x y) = ( x) f ( ) ( y), the is lled strog homomorphism The set ker ={(, ) R R ( )= ( )} is lled the kerel of A ordered set (, f ) is lled epimorphism if : R Rd f : e surjetive d is lled isomorphism if : R R d f : re ijetive Let e equivlee reltio, A d B e two o-empty susets of R We defie ( AB, ) if for every A there exists Bsuh tht (, ) d for every Bthere exists d A suh tht ( d, ) d ( AB, ) if for every Ad B (, ) Let R e - semihyperrig d A equivlee reltio o R is lled regulr if for every x R d, (, ) imply ( x) ( x), ( x) ( x) d ( x) ( x) Let I e o-empty suset of -semihyperrig R We sy tht I is -idel of R if I stisfies the followig oditio: I R I, I R I, R I I for every A -idel I of -semihyperrig R geerte the followig regulr reltio o R : xi y x = y o r x, y I It is esy to see tht I is reflexive, symmetri, trsitive d regulr We shll ll regulr reltio of this type Rees reltio Exmple Let R e -hyperrig d I e orml idel of R The, the reltio x y( modi) is regulr reltio o R Propositio Let R e -semihyperrig d e regulr reltio o R The, R / is - semihyperrig with respet to the followig hyperopertio: ( ) ( )={ ( ) ( ) ( )}, ( ) ˆ ( )={ ( d) d ( ) ( )}, where ={ ˆ } Exmple Let ( R,, ) e semihyperrig suh tht x y = x y x y, e ommuttive group We defie x y = x y for every x, y R d The, R is -semihyperrig Exmple 3 Let ( R,, ) e semirig d (, ) e sugroup of ( R, ) d I e idel of R suh tht I = I ={0} We defie x y = x y I for every x, y R d The, R is multiplitive -semihyperrig Exmple 4 Let R = Z4 d I = ={0,} 4 The, R is multiplitive -semihyperrig with the followig hyperopertio: x ˆ y = {0, }, where x, y R, ˆ d ={ ˆ } Simple semihyperrigs I this setio R is -semihyperrig suh tht it hs elemet 0 with the followig property: x x0, 0 0 = {0} d x 0 = 0 x = {0}, for every x, y Rd Hee {0} is idel of R Defiitio A -semihyperrig R is lled simple (right simple)if () {0} d R re the oly idels (right idels), () RR {0} I the sme wy, we defie simple - hyperrig A idel I of -semihyperrig R is lled simple, if I is simple -semihyperrig This mes tht{0} d I re oly idels of I d II {0} Exmple 5 Let ( F,, ) e field, (, ) e sugroup of ( F, ) suh tht F d { A t } tf e fmily of o-empty disjoit sets i whih A o = The, S = t F A t is simple - semihyperrig with the followig hyperopertios: x y = Ag g, x y = Ag g, where x Ag, y Ag, g, g d

4 IJST (03) 37A3: Exmple 6 Let { A R} e fmily of disjoit set suh tht {0} = 0 A = (0,) 0< < [ mm, ) m < m The, for every x R there exists R suh tht x A So, R is simple R -semihyperrig with the followig hyperopertio: x y = A m, x y = A m, where x A, ya d R m Exmple 7 Let R ={,, } d ={, } The, R is simple -hyperrig with the followig opertios d hyperopertio: where ={, } {,} {,,} Lemm A -semihyperrig R is simple if d oly if R R = R for every R \0 Proof: Suppose tht R is simple - semihyperrig The, R R is idel of R Sie R is simple -semihyperrig, R R is distit from {0}, hee it must e oiide with R, d it follows: RRR = R R = R Let 0 e elemet of R The, R R is idel of R d so either R R = R or R R ={0} If RR ={0}, the the set I = { x R Rx R = {0}}, otis ozero elemet Let x, y I The, R x R ={0} d R y R ={0} Sie R ( x y) R R x RR y R, I is idel of R whih implies tht I ={0} or for I = R If I = R, the R x R = {0} every x R Sie R is simple d {0} RR R R, we hve R R = R But this implies tht R = RR R={0} whih is otrditio Hee, RR = R Coversely, suppose tht R R = R for ll R\0 The, RR {0} If I is idel of R otiig o-zero elemet, the R= RR RIR I, d so I = R Therefore, R is simple -semihyperrig Lemm 3 If I is o-zero miiml idel of -semihyperrig R, the either I I ={0} or I is simple idel of R Proof: Sie I I is idel of R otied i I, we must hve either I I = {0} or I I = I Suppose tht I I = I The, ( I I) I = I I Therefore, III = I I If is o-zero elemet of I, the < > is o-zero idel of R otied i I Sie I is miiml idel of R, we hve I =< > Sie < >= R R R R, we oti II I( RRRR) I II Hee, I I = I d so I is simple idel of R Defiitio 4 Let R e -semihyperrig d for every \0 there exists suh tht for every x R, x x d x x The, R is lled -semihyperrig with -idetity Lemm 5 Let R e right simple - semihyperrig with -idetity The, R \0 is multiplitive lose suset of R Proof: We show tht for every, Rd, R \0 Suppose tht, R\0, ut ={0} Let = { x R x = {0}} The, is right idel of R Sie {0, }, we oti = R whih implies tht R ={0} Hee,

5 55 IJST (03) 37A3: 5-63 ={0} This is otrditio Therefore, R \0 Propositio 6 Let I e simple left idel of - semihyperrig R d for every left idel J of R, y rj implies tht y r J where y I The, for every d r R, I r is either {0} or miiml left idel of R Proof: Suppose tht Ir {0}, where d r R Oviously, Ir is left idel of R I order to show tht it is miiml left idel, let A e left idel of R otied i I r Assume tht ={ x I xr A} The, r A Let x A The, there exists y I suh tht x yr Hee A y r whih implies tht y r A So, r = A Ir Sie I is simple left idel d is left idel of R, the ={0} or = I Hee A ={0} or A = I r Propositio 7 Let I e miiml simple idel of R suh tht L {0} is left idel otied i I So, L L {0} Proof: Sie L R is idel of R otied i I, L R ={0} or L R = I If L R ={0}, the L is idel of R Hee, L = I d II = LI LR = {0} Tht is otrditio Hee, L R = I Sie I = II =( LR) ( LR) ( LL) R We olude tht L L {0} Lemm 8 Let R e multiplitive - semihyperrig with -idetity d I e simple left idel of R The, I = R for every I \0 Proof: Suppose tht 0 is elemet of I The, R is left idel of R otied i I Therefore, R = {0} or R = I If R ={0}, the {0, } is left idel of I Therefore, I I ={0} whih is otrditio Hee, R = I Theorem 9 Let R e simple -hyperrig otiig o-zero miiml left idel The, R is the uio of its miiml left idels Proof: Suppose tht R is simple -hyperrig d I is o-zero miiml left idel The, I Ris idel of R d so either I R ={0} or I R R Suppose tht I R ={0} The, I is idel of R Sie I {0}, it follows tht I = R d so RR R R ={0} It is otrditio We olude tht I R = R d so there exists R suh tht I {0} Let J e o-zero left idel of R otied i I The, ={ I J}, eig o-zero left idel of R otied i I d so J I = J Now, let H = { I R, } The, ertily H is o-zero left idel Let x H, y R d The, there exist, R suh tht x I Hee, x y( I) y = I( y) H Sie R is simple, H = R, d R is the uio of miiml left idels The dul of the previous theorem is true Therefore, if R is simple -hyperrig d otis miiml right idel, the R is the uio of its miiml right idel Propositio 0 Suppose tht R is simple - hyperrig otiig t lest oe miiml left idel d oe miiml right idel The, for every miiml left idel L of R there exists miiml right idel R suh tht L R = R Proof: Let L e miiml left idel of R Sie R is simple -hyperrig, L R ={0} or L R = R By Theorem 9 there exist R d suh tht L {0} Sie {0} L L R, y the dul of the previous theorem there is miiml right idel R suh tht R Sie R is simple -semihyperrig, L R must oiide with R Propositio Let I e proper orml idel of -hyperrig R, A e the set of idels of R otiig I d B e the set of idels of [ R : I ] The, the mp : J [ J : I ] is ilusiopreservig ijetio from A oto B Proof: Sie J is idel of R, [ J : I ] is

6 IJST (03) 37A3: idel of [ R : I ] Hee is well-defied Let J d J e two idels of R suh tht ( J)= ( J ) The, [ J: I ]=[ J : I ] For every elemet x J, there is y J suh tht x y ( modi ) d so x y for some I This implies tht J J I the sme wy J J Hee, J = J Let T e idel of [ R : I ] The, ={ x R I ( x) T} is idel of R d ( )= T Therefore, is ijetive Propositio If I, J re idels of - hyperrig R suh tht I J, I is orml idel d there is o idel K of R suh tht I K J The, [ J : I ] is either simple or ull idel Proof: By Propositio, [ J : I ] is miiml idel of [ R : I ] Thus [ J : I ] is ull or simple, y Lemm 3 3 Noetheri d Artii -semihyperrigs A olletio A of susets of -semihyperrig R stisfies the sedig hi oditio (or A) if there does ot exist properly sedig ifiite hi A A of susets from A Rell tht suset B A is mximl elemet of A if there does ot exist suset i A tht properly otis B Propositio 3 Let R e -semihyperrig The, the followig oditios re equivlet: () R stisfyig the A oditio o right (left) idels () Every o-empty fmily of right (left) idels hs mximl elemet (3) Every right (left) idel is fiitely geerted Defiitio 3 A semihyperrig R is right (left) Noetheri if the equivlet oditios of the ove propositios re stisfied I the sme wy, we defie Artii - semihyperrig Let I e idel of semihyperrig R d I e Noetheri semihyperrig The, I is lled Noetheri idel of R Exmple 8 Let ( R, ) e group d (, ) e sugroup of R The, R is multiplitive Noetheri (Artii) semihyperrig with respet the followig hyperopertio: x y = R Exmple 9 Let A =[, ) for every Z, S A d = Z The, S is Noetheri = Z -semihyperrig ut ot Artii - semihyperrig with respet to the followig hyperopertio: x y = A m, x y = A m, where x A d y A m Propositio 33 Let ( R,, ) e ommuttive rig, (, ) e sugroup of ( R, ) suh tht R d { A g } g R e olletio of o-empty disjoit sets The, S = grag is - semihyperrig with the followig hyperopertio: x y = Ag g, x y = Ag g, where x A g, y A g d Therefore, R is Noetheri (Artii) rig if d oly if S is Noetheri (Artii) -semihyperrig Proof: Oe see tht S is -semihyperrig with the ove hyperopertios Let I e idel of R The, SI = gi Agis idel of R Coversely, suppose tht T is idel of - semihyperrig S The, T = S, where I =< X > d X ={ g R Ag T } Therefore, the ommuttive rig R is Noetheri (Artii) if d oly if S is Noetherio (Artii) Propositio 34 Let I e -idel of - semihyperrig R Let A e the set of idels of R otiig I d B the set of idel of R / I The, the mp : J J / I is ilusio-preservig ijetio of A oto B Proof: The proof is strightforwrd Propositio 35 Let R e Noetheri - semihyperrig d I e -idel of R The, R / I is Noetheri -semihyperrig Proof: The proof is strightforwrd Theorem 36 Let I e Noetheri -idel of - semihyperrig R If R / I is Noetheri - semihyperrig, the R is Noetheri - semihyperrig Proof: Assume tht I d R / I re Noetheri d A A A3 e sedig hi of idels of R There exist sedig hi of idels A I A I, ( AI)/ I ( A I)/ I, i I d R / I, respetively The, there exists N suh tht A I = A I d ( A I)/ = ( A I)/ i I i I I

7 57 IJST (03) 37A3: 5-63 for ll i Hee, A I = A I for ll i i Suppose tht x A I If x I, the i x A I Assume tht x Ai for some i The, there exists x A I suh tht I( x)= I( x ) whih implies tht x = x or x, x I Therefore, x A I whih implies tht A I = A I for ll i Hee, for i i Ai = Ai ( Ai I)= Ai ( A I)= A ( Ai I)= A ( A I)= A So, R is Noetheri respetively The, R = R R is Noetheri (, )-semihyperrig with the followig hyperopertios if d oly if R d R re Noetheri d -semihyperrig, respetively (, ) (, )={( x, y) x, y }, (, ) (, )(, ) = {( x, y) x, } Propositio 37 Let R e -semihyperrig d A, B e two Noetheri -idels of R The, A B is Noetheri su -semihyperrig of R Proof: Sie A d B re -idels, the A B is -idel of A d B is -idel of A B Ideed, ( A B) A( A A) ( B A) A B, ( A B) A ( AA) ( BA) A B I the sme wy we see tht B is -idel of A B We defie : A/ AB ( A B)/ B, y ( AB ( x )) = B ( x ), for ll x A Let AB( x) = A B( y) for some x, y A The, ( x)= ( y) Hee, is well-defied Sie B B ( AB( x) AB( y)) = ({ AB( t) tx y}) ={ B ( t) tx y} = B( x) B( y) = ( ( x)) ( ( y)) AB AB I the sme wy, it is esy to see tht ( ( x) ( y)) = ( ( x)) ( ( y)) AB AB AB A B Hee, A/ AB ( A B)/ B By the previous propositio, A B is Noetheri Lemm 38 Let R e ordered - semihyperrig with zero The priiple idels of R forms hi with respet to ilusio if d oly if idels of R do so Proof: Suppose tht I, J re idels of R suh tht I J with zero The, J I Let x J We osider elemet y I suh tht y J By hypothesis, we hve < x > < y > or < y > < x > If < y > < x >, the y < y > < x > J, whih is impossile Thus we hve < x > < y > d so whih x < x > I Lemm 39 Let R d R e semihyperrig with zero, - d - d - -idetity, Proof: The proof is strightforwrd Propositio 30 Let R e -semirig with zero d -idetity, d e regulr reltios o R suh tht = R R d = Id R The, R / d R / re Noetheri -semirigs if d oly if R is Noetherio -semirig Proof: Let : R R / R / defied y ( x ) = ( ( x ), ( x )) We show tht is homomorphism We hve ( x y ) = ( ( x y ), ( x y )) =( ( x ), ( x )) ( ( y ), ( y )) = ( x ) ( y ) Let f : defied y f ( ) = (, ) I the sme wy, ( x y) = ( x) f ( ) ( y) Also, we hve ker ={(, ) ( )= ( )}={(, ) ( ) = ( ), ( ) = ( )} = Id R Let ( ( x ), ( y )) R / R / There exists R suh tht (, ) d (, ) whih implies tht ( ) = ( ( ), ( )) = ( ( x), ( y)) Hee R R / R / By Lemm 39, R is Noetheri -semihyperrig if d oly if R / d R / re Noetheri -semihyperrigs I the previous propositio, let R e - semihyperrig The, is homomorphism ut is ot strog homomorphism Let R e - semihyperrig A idel P of R is lled prime if AB P, implyig tht A Por B P, where A d B re idels of R A idel M of -

8 IJST (03) 37A3: semihyperrig R is lled mximl, if M R d there re o idels i " etwee" M d R I other words, if I is idel whih otis M s suset, the either I = R or I = M Let R e ommuttive -semihyperrig with zero d - idetity The, every mximl idel of R is prime Exmple 0 Let R ={,,, d }, = d =0, = The, R, is -semihyperrig with the followig hyperopertios: d {,} {,} {,d} {,d} {,} {,} {,d} {,d} {,d} {,d} {,} {,} d {,d} {,d} {,d} {,} d {,} {,} {,} {,} {,} {,} {,} {,} {,} {,} {,d} {,d} d {,} {,} {,d} {,d} For every x, y Rwe defie x y ={, } I this exmple P ={, } is prime idel of R Propositio 3 Let R e ommuttive - semihyperrig with zero d -idetity d M, M,, M differet mximl idels i R The, M, M,, M is proper idel of M M M Proof: It is strightforwrd Propositio 3 Let R e multiplitive - semihyperrig with zero, I d J e o-empty susets of F / d R, respetively The, ) If I is idel of F /, the I R is idel of R ) If J is idel of R, the J F / is idel of F / Proof: () Sie I is o-empty set, there exists =( x, ) I Hee elemet i i i = x (0, ) (, ) (0, ) I for every i i i, whih implies tht I R is o-empty set Now, let x, y I R, the ( x, ), ( y, ) I for every Hee ( x y, ) = ( x, ) ( y, ) I for every d so x y I Let x I R, y R d The, ( x, ) I for every Sie I is idel of F /, ( x, ) ( y, ) I So { ( t, ) t xy} I whih implies tht x y I Hee, I R is idel of R () Let J e idel of R The, 0 J, whih implies tht (0, ) J F / Hee, J F / is o-empty set Let m i = ( xi, i ) d j =( y j, j ) e elemets of J F / The, m i= i i j= j j x xd y x I for every x R m i i j j Hee, i= x x j=y x I whih implies tht m i = x x j = y j jx JF / i i I the sme wy, J F / is losed with respet to the ove hyperopertio Hee, J F / is idel of F / Theorem 33 Let R e multiplitive - semihyperrig with -idetity d zero, I d J e idels of F / d R, respetively The, () ( J ) = I, () ( I ) = J, F / R R F / Proof: () We hve ( JF/ ) R ={ x R ( x, ) JF/ } ={ x R ( x, ) JF /, for every } ={ x R x J, for every, R} Sie R hs -idetity, the ( J F / ) R J Therefore, ( J F / ) R = J () By defiitio, we hve i ( I ) = ( x, ) x x I, for ll x R R F/ i i i i R = i = i i ii i = i = = ( x, ( t, ) I for every t x x, x R This implies tht ( IR ) F / F / I Sie R is with -idetity, ( IR ) F / I Hee, ( IR ) / = I Corollry 34 Let J d J e two idels of R d A d B e two idels of F / suh tht J A J d A B The, ( J) ( J) d R B R F F/ F/ Corollry 35 Let R e multiplitive - semihyperrig The, R is Noetheri (Artii) if d oly if F / is Noetheri (Artii) multiplitive hyperrig Corollry 36 Let R e Artii -rig The, R is Noetheri -rig

9 59 IJST (03) 37A3: Regulr -hyperrigs Let R e -hyperrig A elemet x R is (, )- regulr or shortly regulr if for every x R there exist y R d, suh tht x = x y x Let every elemet of R e regulr The, R is lled regulr -hyperrig A elemet e R is idempotet, if there exists suh tht e = e e I this se, we sy tht e is -idempotet We deote tht the set of ll - idempotet elemet with E Hee, if E is set of ll idempotet elemets of R, the E = E If every elemet of R is idempotet, the R is lled idempotet -hyperrig For elemet i -hyperrig R if there exist elemet R d, suh tht = d =, the is sid to e (, )- iverse of I this se, we write V ( ) Notie tht elemet with iverse is eessry regulr d every (, )-regulr elemet hs (, ) -iverse: if there exist x R d,, = x, the defie = x x d oserve tht = ( xx) =( x) x = x= =( x x) ( x x)= x ( x) ( x x) = x( x) x = xx = Hee, i regulr -hyperrig every elemet hs iverse Let e (, )-regulr elemet of R The, V ( ) A idel I of - hyperrig R is lled (, )-regulr, if x ( x xyx) I implies tht there exist x, x x x y x suh tht x = x xfor some I Let I e idel of -hyperrig R d for every x I there exist, d y I suh tht x = xy x The, I is lled regulr idel of R Exmple Let ( R,, ) e regulr ommuttive rig, (, ) e o-empty suset of R d e equivlee reltio defied s follows: x y x = y or x = y The, the set R / ={ ( x) x R} eomes regulr -hyperrig with respet to the hyperopertio ( x ) ( y)={ ( x y), ( x y)} d multiplitio ( x ) ( y)= ( x y) Propositio 4 Let R e regulr -hyperrig with zero The, every priipl (right) left idel of R is geerted y idempotet elemet Proof: Suppose tht x is elemet of R The, there exist y R d, suh tht x = xyx Sie, m < x > l = y R y ix rjjx, i,, m, j j, rj R, j m i= j= We hve < x > l =< y x > l Moreover, ( y x) ( yx)= y( xyx)= y x Hee, y x is -idempotet elemet of R This ompletes the proof Propositio 4 Let R e -hyperrig with - idetity d for every x R there exists idempotet elemet er suh tht x R = er The, R is regulr -hyperrig Proof: Sie R is -hyperrig with -idetity, there exist x, x R d suh, tht x = ex d e = xx Sie e is idempotet elemet of R, there exists suh tht e = e e Hee, x = ex =( ee) x = ( xx ) ( xx ) x = xx ( xxx) = xx x The, x is regulr elemet of R Therefore, R is regulr -hyperrig Propositio 43 Let R e regulr -hyperrig The, every oe-sided idel of R is idempotet Proof: Suppose tht I is right idel of R d x I The, x = xyx for some, d y R Cosequetly, x =( x y) x II Thus, I I = I Propositio 44 Let I, J e two idels i - hyperrig R suh tht I J If J is regulr, the I is regulr too Proof: Suppose tht x I The, there exist y J d, suh tht x = xy x Hee, z = yx y is elemet of I suh tht x zx = x( yxy) x = ( xyx) ( yx)= xyx = x Propositio 45 Let R e -hyperrig d I J re idels of R suh tht I is (, )- regulr idel of R The, I d [ J : I ] re oth regulr if d oly if J is regulr Proof: By Propositio 44, it is ovious tht if J is

10 IJST (03) 37A3: regulr, the [ J : I ] d I re regulr Coversely, ssume tht I d [ J : I ] re oth regulr d x J It follows from regulrity [ J : I ] tht ( x x y x) I for some, d y J Suppose tht x ( x xyx) I The, there exist x, x x x y x suh tht some J Hee x = x x for x x x yx = xx xyx ( x xyx) ( x xyx) xyx = x x xxyx xyxx xyxxyx = x( xy yx yxxy y) x Cosequetly, from whih we olude tht x = xzx from some z J Hee, J is regulr Propositio 46 Let R e -hyperrig, I e regulr idel d J e orml idel suh tht (, ) -regulr The, I J is regulr idel of R Proof: Clerly, J is orml idel of I J d I J is orml idel of I Cosequetly, oe see [ I J : J ] [ I :( I J) ] Sie I is regulr, [ I :( I J) ] is regulr By Propositio 45, we oti I J is regulr idel of R Let R e -rig suh tht every elemet of R e (, )-regulr (This mes tht for every x R there exists y Rd suh tht x = x y x ) The, R is lled (, ) -regulr -rig A idel P of ommuttive -rig R is lled qusi prime if ARB P implies tht A Por B P, where A, B re susets of R d R A A, RB B Theorem 47 Let R e ommuttive -rig d suh tht () R e Noetheri rig, () R / P is (, ) -regulr for ll qusi prime idels of R, (3) {0} is qusi prime idel of R The, R is (, ) -regulr -rig Proof: Assume tht R is ot (, )-regulr - hyperrig The, there is some x R suh tht x xrx Note tht {0} is qusi prime idel of R suh tht xx R {0} Hee, ={ I R x xrx I, I is qusi idel of R} is o-empty set Sie R is Noetheri -rig, there is qusi idel J i R whih is mximl with reset to the property x x R x J Suppose tht x J e (, ) -regulr elemet of R/ J Thus, there is y R suh tht x x yxj xrx J whih is otrditio Hee, R / J is ot (, ) -regulr Therefore, J is ot qusi prime idel of R Thus there exist idels I suh tht I RI J d I, I re ot susets of Now, set A ={ r R rri J} d B ={ rr ARr J} Sie J is qusi prime idel, we olude tht A d B re qusi prime idels of R Clerly, I A d of mximlity of, I B Hee, A d B properly oti J Beuse J x R x A d x x R x B Hee, there exist r, r R d, suh tht x = x r x d x = xr x where Ad B We hve x x ( rr rxr) x=( xxrx) ( xxrx) rxa I the sme wy, we see x x ( rr rxr) x B Hee, x x R x A B Note lso tht, sie ( A B) R( AB) ARB J, the we hve A B J Hee, xx R x J, whih is otrditio Therefore, R is (, ) - regulr -rig Propositio 48 Let R e ommuttive - hyperrig with -idetity The, IJ = I J for ll idels I d J if d oly if R is regulr Proof: Suppose tht R is regulr d I, J re two idels of R By Propositio 43, I J is idempotet The, I J =( I J) ( I J) I J Hee, IJ = I J Coversely, suppose tht IJ = IJ, where I d J re idels of R Let x R The, x R d R x re idels of R So, ( x R) ( Rx)= ( xr) ( Rx) Sie R hs - idetity, x x R d x R x d so x ( x R) ( Rx)=( xr) ( Rx) xrx Hee, R is regulr Propositio 49 Let e regulr reltio o regulr - hyperrig R, ( ) is -idempotet elemet i R / d e (, )-regulr elemet of R The, there exists idempotet elemet e R suh tht ( )= ( e ), where e is -idempotet elemet of R Proof: It is strightforwrd Propositio 40 Let R e regulr -hyperrig d e e e (, )-regulr elemet of R, where

11 6 IJST (03) 37A3: 5-63 d, e E, e E The, E ( e, e) ={ x V ( ee) E xe = x e = x, ex = ex = x}, is o-empty set of R Proof: Sie e e is (, )-regulr, V ( e e) is o-empty set of R Suppose tht yv ( ee ) d x = e y e The, ( ee ) x ( ee ) = ( ee ) ( eye ) ( ee ) = e ( ee ) y ( ee ) e = ( ee ) y ( ee ) = ee Also, we hve x ( e e) x =( e ye) ( ee) ( e ye) = ey( ee) ( ee) ye = eyeeye = e( yeey) e = eye = x, d so x V ( e e) Also, x x =( e ye) ( ey e) = e( y( ee) y) e = eye = x, d so x E Filly, it is ler tht x e = x d e x= x Therefore, x E ( e, e) whih implies tht E ( e, e) is o-empty set of R Propositio 4 Let R e regulr -hyperrig d e e e (, )-regulr elemet of R, where, d e E, e E The, E ( e, e ) ={ x E xe = e x = x, exe = ee } Proof: Suppose tht x E, x e = e x = xd e xe = e e The, ( ee) x( ee) =( ee) ( xe) e = e( ex) e = exe = e e x( ee) x = ( xe) ex = x x = x Hee, x V ( e e) d so x E ( e, e) Coversely, let x E ( e, e) The, x E d ( ee ) x( ee ) = e e Moreover, ( e e ) x( e e ) = ( e e ) ( xe ) e =( e e ) xe = e ( e x) e = e xe Therefore, e xe = e e Propositio 4 Let R e regulr -hyperrig, e ee (, )-regulr elemet of R, where e E, e E The, E ( e, e) is su semigroup of Proof: By Propositio 4, E ( e, e) is oempty set Let x, x E ( e, e) The, x, x E d x e = e x = x, e x e = e e, x e = e x = x, e x e = e e Moreover, x xx =( xe) x( ex) = x( exe) x = x( ee) x =( xe) ( ex) = x x = x It follows tht ( x x) ( xx) = ( xxx) x = x x, d so x x is -idempotet elemet Also, ( x x) e = x( xe ) = x x, e ( xx) = ( ex) x = x x e ( xx ) e =( ex) ( e x ) e =( exe) x e =( ee) x e = e( ex ) e = e x e, d so x x E ( e, e) This implies tht E ( e, e ) is su semigroup of R Propositio 43 Let R e regulr -hyperrig, is (, ) -regulr, is (, )-regulr d for every -idempotet elemets e, e R, e e E ( e, e ) The, V ( ) V ( ) V ( ) V Proof: Sie is (, ) -regulr d is (, ) -regulr elemets, V ( ) d () re o

12 IJST (03) 37A3: empty susets of R Suppose tht V ( ) d V ( ) Let g E (, ) The, ( ) ( g ) ( ) = ( g) ( ) = ( g) = g =( ) g( ) = ( g ) = ( ) =( ) ( ) = ( g ) ( ) ( g )= ( g) ( g) = ( gg) = g Hee, g V ( ) Oe see tht d re -idempotet elemets of R Hee ( ) ( ) E (, ) This implies tht (( ) ( )) V ( ) Thus, ( ) ( ) V ( ) whih implies tht V ( ) Therefore, V () V () V ( ) Propositio 44 Let R e regulr -hyperrig with the set E of -idempotet elemets, e e is (, )-regulr elemet of R where e, e E d V () e E for every e E The, E is su semigroup of R Proof: Suppose tht e, e E, x V ( e e) d y = e x e The, ( ee) y( ee) =( ee) ( exe) ( ee) = e( ee) x( ee) e =( ee) x( ee) = e e y ( ee) y = ( exe) ( ee) ( exe) = ex( ee) ( ee) xe = e( xeex) e = e xe Hee, y V ( e e) Moreover, y y =( exe) ( e xe) = e ( xe e x ) e = e xe = y, whih implies tht y is -idempotet elemet But e e V ( y ) E Hee, E is su semigroup of R 5 Colusio I this work, we preseted the oept of - semihyperrig whih is ew kid of hyperlger d is geerliztio of semihyperrigs, hyperrigs d rigs d proved some results I prtiulr, the study of the otios of Noetheri, Artii, simple d regulr -semihyperrigs Referees [] Nousw, N (964) O geerliztio of the rig theory Osk J Mth, [] Bres, W E (966) O the -rigs of Nousw Pifi J Mth, 8(3), 4-4 [3] Luh, J (969) O the theory of simple -rigs Mihig Mth J, 6, [4] Kyuo, S (978) O prime -rigs Pifi J Mth, 75(), [5] Mrty, F (934) Sur ue g e erliztio de l otio de groupe 8th Cogress Mth Sdeves, Stokholm, [6] Corsii, P (993) Prolegome of hypergroup theory Seod editio, Avii editor [7] Vougiouklis, T (994) Hyperstrutures d their represettios Florid, Hdroi Press [8] Preowitz, W & Jtosik J (979) Joi Geometries Spriger-Verlg, UTM [9] Dvvz, B & Leoreu-Fote, V (007) Hyperrig Theory d Applitios USA: Itertiol Ademi Press [0] Bosig, P & Corsii, P (98) O semihypergroup d hypergroup homomorphisms Boll U Mt Itl B, 6(), [] Corsii, P (980) Sur les semi-hypergroupes Att i So Pelori-t Si Fis Mt Ntur, 6, [] Corsii, P & Leoreu, V (003) Applitios of hyperstruture theory Adves i Mthemtis, Kluwer Ademi Pulishers, Dordreht [3] Dvvz, B (000) Some results o ogruees o semihypergroups Bull Mlys Mth Si So (), 3, [4] Dvvz, B & Pourslvti, N S Semihypergroups d S-hypersystems, Pure Mth Appl, (000), [5] Fsio, D & Frei, D (007) Existee of proper semihypergroups of type U o the right Disrete Mth, 307, [6] Gut, C (99) Simplifile semihypergroups Algeri hyperstrutures d pplitios (Xthi, 990), 03-, World Si Pul, Teek, NJ [7] Leoreu, V (000) Aout the simplifile yli semihypergroups Itertiol Alger Coferee (Isi, 998), Itl J Pure Appl Mth, 7, [8] Rot, R (98) Sugli iperelli moltiplitivi Red Di Mt, Series (4), 7-74 [9] Ro, M K (995) -semirig Southest Asi Bulleti of Mths, 9, [0] Dutt, T K & Srdr, S K (00) O the opertor semirigs of -semirig Southest Asi Bull

13 63 IJST (03) 37A3: 5-63 Mth, 6, 03-3 [] Avriyeh, S M, Mirvkili, S & Dvvz, B (00) O -hyperidels i -semihypergroups Crpthi J Mth, 6(), -3 [] Heidri, D, Dehkordi, S O & Dvvz, B (00) -Semihypergroups d their properties UPB Si Bull, Series A, 7, 97-0