Itratial Jural f athmatial Arhiv-7(5), 06, 60-70 Availabl li thrugh wwwimaif ISSN 9 5046 IDEALS IN ALOST SEILATTICE G NANAJI RAO, TEREFE GETACHEW BEYENE*,Dpartmt f athmatis, Adhra Uivrsity, Visakhpataam, 530003, Idia E-mail: ai6us@yahm, drgaairamath@auvspdui Addis Ababa Si ad Thlgy Uivrsity, Addis Ababa, Ethipia E-mail: gtrf4@gmailm (Rivd O: 8-04-6; Rvisd & Aptd O: 8-05-6) ABSTRACT Th pt f idal i a Almst Smilatti (ASL) is itrdud ad th smallst idal taiig a giv mpty subst f a ASL L is dsribd Als, svral prprtis f idals ar drivd Prvd that th st I (L), f all idals i a ASL L, is a distributiv latti ad als, th st PI (L), f all priipal idals frm smilatti is stablishd Drivd st f quivalt idtitis fr th itrsti f ay family f idals is agai a idal ad a - rrspd btw idals (prim) f L ad idals (prim) PI (L) is stablishd Fially btaid, vry amiabl st i L is mbddd i a smilatti PI (L) Ky Wrds: Idal, Priipal Idal, Prim Idal, Distributiv Latti, Cmplt Latti, Amiabl St AS Subt lassifiati (000): 06A99, 06D0 INTRODUCTION Idals wr first studid by Ddkid, wh dfid th pt fr rigs f algbrai itgrs Latr th pt f idal was xtdd t rigs i gral H St ivstigatd idals i Bla rigs, whih ar latti f a spial kid Thr is alrady a wll-dvlpd thry f idals i latti W wish t shw that it is usful t xtd th ti f idal t th mr gral systms alld Almst Smilatti Thr ar ly rasabl way f dfiig what is t b mat by a idal i a latti Rall that Ddkid s dfiiti f a idal i a rig R is that it is a llti J f lmts f R whih () tais th diffr a b, ad h th sum a + b, f ay tw f its lmts a ad b fr all a, b J, ad () tais all multipls suh as ax r ya f ay f x, y R ad a J, By aalgy, a llti J f lmts f a latti L is alld a idal if () it tais th latti sum a b f ay tw f its lmts a ad b, ad () it tais all multipls a x f ay x L ad a J Th aalgy is that th gratst lwr bud, r latti mt a b rrspds t prdut i a rig, ad th last uppr bud, r latti i a b rrspds t th sum f tw lmts i a rig I this papr, w itrdu th pt f a idal ad smallst idal taiig a giv mpty st i a ASLL with biary prati ad prv that th st I (L) f all idals f L frms a distributiv latti, th st PI (L), f all priipal idals f a ASL L is a smilatti Als, giv a quivalt ditis fr th itrstis f arbitrary family f idals i a ASL L is agai a idal W stablish a -t- rrspd btw idals f L ad idals f PI (L), i partiular a -t- rrspd btw prim idal f L ad prim idal f PI (L) I this papr, w prv that if I is a idal f L ad K b a mpty subst f L whih is lsd udr th prati ad I K =, th thr xists a prim idal P f L suh that I P ad P K = Fially, w btai vry amiabl st is mbddd i a smilatti PI (L) Crrspdig Authr: Trf Gtahw By* Addis Ababa Si ad Thlgy Uivrsity, Addis Ababa, Ethipia Itratial Jural f athmatial Arhiv- 7(5), ay 06 60
G Naai Ra, Trf Gtahw By* / Idals i Almst Smilatti / IJA- 7(5), ay-06 I sti, w llt a fw imprtat dfiitis ad rsults whih ar alrady kw ad whih will b usd mr frqutly i th papr I sti 3, w itrdu th pt f idal i a ASL L ad prv that th st I (L) f all idals f L frms a distributiv latti fr whih th st PI (L) f all priipal idals f L is a smilatti I this sti, w driv st f idtitis fr th itrsti f arbitrary family f idals i L is agai a idal ad h th st I (L) is a mplt latti I sti 4, w dfi a prim idal i a almst smilatti ad prv that a -t- rrspd btw I (L) ad th st f all idals i PI (L) als, prv that a -t- rrspd btw th st f all prim idals i L ad th st f all prim idals i PI (L) Fially, w prv that vry amiabl st i L is mbddd i th smilatti PI (L) PRELIINARY I this sti w llt a fw imprtat dfiitis ad rsults whih ar alrady kw ad whih will b usd mr frqutly i th papr Dfiiti : [] A smilatti is a algbra (, ) S satisfyig: x ( y z) = ( x y) z x y = y x 3 x x= x, fr all x, y, z S S satisfyig whr S is mpty st ad is a biary prati Dfiiti : A idal i a smilatti L is a mpty subst whih lsd udr iitial sgmts Dfiiti 3: A prpr idal P f a smilatti ( L, ) is said t b prim idal if fr ay a, b L, a b P, th ithr a P r b P I thr wrds, a smilatti is a idmptt mmutativ smigrup Th symbl a b rplad by ay biary prati symbl, ad i fat w us f th symbls f,, + r, dpdig th sttig Th mst atural xampl f a smilatti is ( P ( X ), ), r mr grally ay llti f substs f X lsd udr itrsti A sub smilatti f a smilatti ( S, ) is a subst f S whih is lsd udr th prati A hmmrphism btw tw smilattis ( S, ) ad ( T, ) is a map h : S T with th prprty that h( x y) = h( x) h( y) far all x, y S A ismrphism is a hmmrphism that is - ad t It is wrth thig that, baus th prati is dtrmid by th rdr ad vi vrsaals, it a b asily bsrvd that tw smilattis ar ismrphi if ad ly if thy ar ismrphi as rdrd sts Dfiiti 4: [3] A algbra ( L, ) f typ () is alld a Almst Smilatti if it satisfis th fllwig axims: ( AS) ( x y) z = x ( y z) (Assiativ Law) ( AS ) ( x y) z = ( y x) z (Almst Cmmutativ Law) ( AS 3) x x = x (Idmptt Law) Dfiiti 5: [3] Lt L b a mpty st Dfi a biary prati L by: x y = y, fr all x, y L Th ( L, ) is alld disrt ASL Dfiiti 6: [3] Fr ay if a b = a a, b L whr L is a ASL, w say that a is lss r qual t b ad writ a b, Dfiiti 7: [3] Lt L b a ASL Th fr ay a, b L, w say that a is mpatibl with b ad writ a ~ b if ad ly if a b = b a A subst S f L is said t b mpatibl st if a ~ b, fr all a, b S Dfiiti 8: [3] Lt L b a ASL Th a maximal mpatibl st i L is alld a maximal st Dfiiti 9: [3] Lt b a maximal st i L Th a lmt xists a suh that a x = x x L is said t b amiabl if thr 06, IJA All Rights Rsrvd 6
G Naai Ra, Trf Gtahw By* / Idals i Almst Smilatti / IJA- 7(5), ay-06 Lmma 0: [3] Lt L b a ASL Th fr ay a, b L, a b = b a whvr a b Thrm : [3] Lt b a maximal st i L ad a Th fr ay x L x a, x L is -amiabl, th thr is a smallst lmt a with a x = x W dt this lmt a f L by Crllary : [3] If is a maximal st ad th prprty x Crllary 3: [3] Lt b a maximal st ad suh that x x = x ad x x = x x L b -amiabl Th x is th uiqu lmt f Dfiiti 4: [3] If is a maximal st i L, th w dt th st f all -amiabl lmts f L by A (L) Thrm 5: [3] Lt b a maximal st Th ( A, ) is a ASL rvr, fr ay x, y A, w hav ( x y) = x y Dfiiti 6: [3] A maximal st i L is said t b amiabl if A = L That is, vry lmt i L is -amiabl Dfiiti 7: [3] A lmt m L is said t b uimaximal if m x = x fr all x L Dfiiti 8: If ( P, ) is a pst whih is budd abv i whih vry mpty subst f P has glb, th vry mpty subst f P has lub ad h is a mplt latti 3IDEALS ASL L I this sti, w itrdu th pt f a idal i a ad dsrib th smallst idal taiig a giv mpty subst f L W furthr prv that th st I (L) f all idals f L frms a distributiv latti fr whih th st PI (L) f all priipal idals f L is a smilatti Thrughut th rmaiig f this sti, by L w ma a ASL ( L, ) ulss thrwis spifid I th fllwig, w giv th dfiiti f a idal i a ASL L Dfiiti 3: A mpty subst I f a ASL L is said t b a idal if x I ad a L, th x a I Frm th dfiiti f idal i ASL L, it a b asily s that vry idal is lsd udr th prati ad h vry idal is a sub ASL f L But, ay subst f L whih is lsd udr th prati d t b a idal Fr, sidr th fllwig xampl: Exampl 3: Lt L = { a, b, } Dfi a biary prati L as blw: a b a a a a b a b a b I this ASL, th st { a, b} is lsd udr th prati, but t a idal, si b = { a, b} I th fllwig thrm, w dsrib th idal gratd by a giv mpty subst S f L ; that is, th smallst idal f L taiig S Thrm 3: Lt S b a mpty subst f L Th ( S] = {( i= si) x x L, si S whr i ad is a psitiv itgr} is th smallst idal f L taiig S 06, IJA All Rights Rsrvd 6
G Naai Ra, Trf Gtahw By* / Idals i Almst Smilatti / IJA- 7(5), ay-06 Prf: Supps S is a mpty subst f L Th fr ay s S, w hav s = s s ad h S (S] Thus (S] is mpty W shall prv that (S] is a idal Lt a (S] ad t L Th a = ( i= si ) x fr sm x L ad s i S fr all i Nw, a t = ( i= si ) x) t = ( i= si ) ( x t) = ( i= si ) y whr y = x t L Thus a t (S] H (S] is a idal f L Nw, it rmais t prv that (S] is th smallst idal f L taiig S Supps J is a idal f L suh that S J Th fr ay t (S], w hav t = ( i= si ) x fr sm x L ad s i S whr i This implis that t = ( i= si ) x, x L ad S J fr all i Thus t J H ( S] J Thrfr (S] is th smallst idal taiig S s i If S = { a}, th w writ (a] istad f (S] ad is alld th priipal idal f L gratd by a Nw, w hav th fllwig Crllary 33: Lt L b a ASL ad a L Th = { a x x L} is a idal f L Crllary 34: Fr ay a, b L, a ( b] if ad ly if a = b a Prf: Supps a (b] Th a b t Thrfr = fr sm t L Nw, b a = b ( b t) = ( b b) t = a = b a Cvrs fllws by th dfiiti f (a] b t = a Crllary 35: Lt I b a idal f L Th, fr ay a, b L, a b I if ad ly if b a I Prf: Supps I is a idal f L ad supps that a b I Th ( a b) a I It fllws that b a = b ( a a) = ( b a) a = ( a b) a I Similarly, w a prv th vrs Crllary 36: Fr ay a, b L,( a b] = ( b a] Prf: Si ( a b) t = ( b a) t fr all t L, it fllws that ( a b] = ( b a] Rall that fr ay a, b L, with a b, w hav a b = b a Nw, w hav th fllwig Crllary 37: Lt I b a idal f L Th, fr ay x I ad a L, a x I ad h I is a iitial sgmt f L ; that is, x I ad a L suh that a x imply that a I Prf: Supps I is a idal f L ad supps fllws that a I, si x a I x I ad a L suh that x a Th a = a x = x a It It is lar that vry iitial sgmt I f L tais th zr lmt 0 Nw, w hav th fllwig thrm Thrm 38: Lt L b a ASL Th th itrsti f ay lass f a iitial sgmts f L is als a iitial sgmt f L Prf: Lt { I } b a lass f a iitial sgmts f L If th idx st J is mpty, th = L, whih is I larly a iitial sgmt f L Supps J is mpty Si ah iitial sgmt tais 0, it fllws that tais 0 Nw, w shall prv that I is a iitial sgmt f L Lt a x Th x I fr all J ad a x Si I is a iitial sgmt f L, Thrfr a I Thus I is a iitial sgmt f L a L ad x I I suh that a I fr all J 06, IJA All Rights Rsrvd 63
G Naai Ra, Trf Gtahw By* / Idals i Almst Smilatti / IJA- 7(5), ay-06 Thrm 39: Lt sgmt f L { I } b a mpty lass f a iitials sgmt f a ASL L Th I is als a iitial Prf: Put I = I Si th idx st is mpty ad ah I is mpty, it fllws that I is mpty Supps a L ad x I with a x Th x I fr sm J Si I is a iitial sgmt f L, x I fr sm J Thrfr x I Thus I is a iitial sgmt f L Nt that, I (L) dt th st f all iitial sgmt f a ASL L Thrm 30: I (L) is a mplt latti, with rspt t st ilusi, i whih fr ay { I }, glb { I J } J = I ad lub { I } J = I I th fllwig, w prv that, th st I (L) f all idals i a ASLL is a distributiv latti Fr this, first w d th fllwig Dfiiti 3: Lt I ad J ar idals i L Th I J = { x L x I ad x J} Lmma 3: If I ad J ar idals f a ASL L, th I J is a idal f L Prf: Supps I ad J ar a idals f L Th larly I ad J ar mpty substs f L H w a hs x I ad y J Th w hav x y I ad h y x I Als, w hav y x J Thrfr y x I J H I J is mpty Clarly I J is a idal f L It a b asily s that I J = { a b a I ad b J} Dfiiti 33: Lt I ad J ar idals i L Th I J = { x L x I r x J} Lmma 34: If I ad J ar idals f L, th I J is a idal f L Prf: Supps I ad J ar idals f L Th larly I ad J ar mpty substs f L H I J is mpty W shall prv that I J is a idal f L Lt x I J ad a L Th ithr x I r x J If x I, th x a I I J Thus I J is a idal f L It a b asily s that th itrsti (ui) f ay fiit family f idals f a hav th fllwig thrm whs prf is straightfrward ASL L is agai a idal Nw, w Thrm 35: Th st I (L) f all idals f a ASL L is a distributiv latti with rspt t st ilusi Nxt, w prv that th st PI (L), f all priipal idals i a fllwig ASL L is a smilatti Fr this, w d th Lmma 36: Fr ay a, b L, b (a] if ad ly if ( b] Prf: Supps b (a] Th b a b Thus ( b] Cvrs is trivial, si a (b] = Nw, lt t (b] Th t = b t = ( a b) t = a ( b t) (a] Lmma 37: Lt a, b L Th ( b] whvr a b a b Th a = a b Nw, lt t (a] Th t = a t = ( a b) t = ( b a) t = Prf: Supps b ( a t) ( b] Thrfr ( b] 06, IJA All Rights Rsrvd 64
G Naai Ra, Trf Gtahw By* / Idals i Almst Smilatti / IJA- 7(5), ay-06 Lmma 38: Fr ay a, b L, ( a b] ( b] = ( b a] Prf: Supps a, b L ad supps t ( b] ( b] Th t (a] ad t (b] Thus w hav t = a t ad t = b t Nw, t = a t = a ( b t) = ( a b) t ( a b] H ( b] ( a b] Cvrsly, lt t ( a b] Th t = ( a b) t = a ( b t) ( a] ad als, t = ( a b) t = ( b a) t = b ( a t) ( b] Thus t ( b] H ( a b] ( b] ( b] Thrfr ( b] = ( a b] H ( a b] ( b] ( b] = ( b] = ( b a] Nw, w hav th fllwig thrm, whs prf fllws by th abv lmmas Thrm 39: Lt L b a ASL Th th st PI (L) f all priipal idals f L is a smilatti It a b asily s that th abv smilatti PI (L) is t a sub-latti f th distributiv latti I (L) Lt us rall that a lmt a f L is said t b miimal if x L, x a imply that x = a Obsrv that, fr ay a L, a is miimal if ad ly if b a = a fr all b L Als, bsrv that L has 0 if ad ly if L has uiqu miimal lmt Als s that, if x is a miimal lmt i L ad I is a idal f L, th x I W hav bsrvd that th itrsti f a fiit family f idals is agai a idal But, th itrsti f a arbitrary family f idals d t b a idal agai, i a gral ASL I th fllwig thrm w giv st f idtitis fr th itrsti f a arbitrary family f idals is agai a idal Fr this, first w d th fllwig Lmma 30: If L has miimal lmt, th th st f all miimal lmts f L frms a idal Prf: Supps L has a miimal lmt say a Nw, put I = { m m is a miimal lmt i L} a I Lt x I ad t L Th s x = x fr all s L I is mpty, si Nw, s ( x t) = ( s x) t = x t fr all L Thrfr I is a idal ASL L () Th itrstis f ay family f idals is mpty () Th itrstis f ay family f idals is agai a idal (3) Th lass I (L) has last lmt (4) Th lass I (L) is mplt (5) Th lass PI (L) has last lmt (6) L has a miimal lmt Thrm 3: Th fllwig ditis ar quivalt i a Prf: Supps Th larly s Thus x t is a miimal lmt f L H x t I {I α} α δ b a family f idals i L ad supps I Iα α δ 06, IJA All Rights Rsrvd 65 = is mpty Th larly I is mpty ad I is a idal f L This prvs () () If w tak I = Iα Th by (), I is a idal f I α I L, ad larly I is th last lmt f I (L) This prvs () (3) Si I (L) is budd abv by L ad ay mpty subst f I (L) has glb, fllws by I (L) has last lmt H (3) (4) Clarly (4) (5) (5) (6) : Supps PI (L) has last lmt say (a] Nw, w shall prv that a is a miimal lmt i L Supps x L suh that x a Th w hav ( x] It fllws that ( x ], si (a] is miimal H a = ( x] Thrfr a = x a = a x, si x a H a x Thrfr by atisymmtri, x = a Thus a is miimal (6) () fllws by vry idal tais a miimal lmt 4THE SEILATTICE PI (L) W hav prvd i th prvius sti that th lass PI (L) f all priipal idals f L frms a smilatti I this sti, w prv that a -t- rrspd btw st f all idals (prim idals) i L ad st f all idals (prim idals) i PI (L)
G Naai Ra, Trf Gtahw By* / Idals i Almst Smilatti / IJA- 7(5), ay-06 Thrughut th rmaiig f this sti, by L w ma a ASL ( L, ) ulss thrwis spifid I viw f rlary 35, w giv th fllwig dfiiti f a prim idal i a ASL L whih iids with th wll kw pts f prim idal i smilatti Dfiiti 4: A prpr idal P f L is said t b x P r y P a prim idal if fr ay x, y L, x y P implis that Lmma 4: A prpr idal P f L is prim if ad ly if fr ay idals I ad J f L, I P r J P I J P implis that Prf: Supps I ad J ar a idals f L suh that I J P If I P Assum that I P Th thr xists x I suh that x P Lt y J Th y x J x y J Als, x y I H x y I J P Si P is prim idal ad x P, y P J P Cvrsly, assum th diti W shall prv that P is a prim idal Lt x y L x y P Th ( x] ( y] = ( x y] P Thrfr ( x] P r ( y] P H x P r y P P is prim, th th lmma hlds tru ad h Thus, suh that Thrfr Th fllwig thrm stablishs th rlati btw th idals (prim idals) f L ad th idals (prim idals) f th smilatti PI (L) Thrm 43: Lt L b a ASL Th w hav th fllwig: Fr ay idal I f L, I := { a I} is a idal f PI (L) rvr, I is prim if ad ly if I is prim Fr ay idal K f th smilatti PI (L), K = { a L K} is a idal f L Furthr, K is prim if ad ly if s is K 3 Fr ay idals I ad I f L, I I if ad ly if I I 4 Fr ay idals K ad K f PI (L), K K if ad ly if K K 5 I = I fr all idals I f L 6 K = K fr all idals K f PI (L) Prf: Supps I is a idal f L Th = { a I} Si I is mpty, it fllws that I is mpty Lt I Nw, w shall prv that I is a idal f PI (L) I ad (t] PI (L) Th a I ad t L Thrfr a t I H ( t] = ( a t] I Thus I is a idal f PI (L) Supps I is a prim idal f L W shall prv that I is a prim idal f PI (L) Lt ( a ],( b] PI (L) suh that ( b] I Th ( a b] I Thrfr ( a b] = ( t] fr sm t I Si a b ( a b] = ( t], a b = t ( a b) Thrfr a b I Si I is prim, ithr a I r b I It fllws that I r ( b] I Thus I is a prim idal f PI (L) Cvrsly, supps I is a prim idal f PI (L) Lt a, b L suh that a b I Th ( b] = ( a b] I Thrfr I r ( b] I H ( a ] = ( s] r ( b ] = ( t] fr sm s, t I Thrfr a = s a I r b = t b I ad h I is prim Supps K is a idal f PI (L) Th K = { a L K} W shall prv that K is a idal f L Si K is mpty, K is mpty Lt a K ad t L Th K ad (t] PI (L) Thrfr ( a t] ( t] K H a t K Thus K is a idal f L Nw, supps K is a prim idal f PI (L) W shall prv that K is a prim idal f L Lt a, b L suh that a b K Th ( b] = ( a b] K Thrfr ithr K r ( b] K, si K is prim It fllws that 06, IJA All Rights Rsrvd 66
G Naai Ra, Trf Gtahw By* / Idals i Almst Smilatti / IJA- 7(5), ay-06 a K r Nw, lt b K H K is a prim idal f L Cvrsly, supps ( a ],( b] PI (L) suh that ( b] K Th ( a b] K It fllws that K is prim, ithr PI (L) a K 3 Supps I ad Thrfr ( ] I Thus Thrfr ] 4 Supps K ad r b K Thrfr a] K K is a prim idal f L a b K Si ( r ( b] K H K is a prim idal f I ar idals f L suh that I I Lt I Th a I ad h a I a I I Cvrsly, supps I I Lt a I Th I I ( a ] = ( t fr sm t I H a = t a I Thus I I K Lt K K Thus K ar a idals f PI (L) suh that K a Th ( a ] = ( t fr sm t K H a = t a K Thus K K Cvrsly, K Lt K Th a K K Thus a K H K Thrfr K Thrfr ] supps K K 5 Supps I I K a I Th Clarly I Thrfr ( ] = ( t] I I Thus a fr sm t I H a = t a I Thrfr I = I Similarly, w a prv (6) Lmma 44: Lt I ad J b a idals f L Th ( I J ) = I J Prf: Supps I ad J ar a idals f L Th ( I J ) I, J H ( I J ) I J Cvrsly, supps I J I, J Thrfr by thrm 43, w hav I J Th I ad J H ( a ] = ( t] fr sm t I ad ( a ] = ( s] fr sm s J Thrfr a = ( t] ad h a = t a Similarly w gt a = s a Si t I, a = t a I Similarly w gt a J H a I J It fllws that ( a ] ( I J ) Thrfr I J ( I J ) ad h ( I J ) = I J Thus w hav th fllwig thrm, whs prf fllws by thrm 43 ad lmma 44 Thrm 45: Th mappig I I is a -t- rrspd f I (L) t I (PI (L)) rvr, this rrspd givs - t - rrspd btw th prim idals f L ad ths f PI (L) Nw, w prv th fllwig thrm Thrm 46: Lt L b a ASL with a miimal lmt ad lt tais prisly th miimal lmts f L dt th last lmt f I (L) Th Prf: Supps x W shall prv that x is miimal Supps a L suh that a x Th by lmma 37 ( x] ad ( x] O th thr had, is th last lmt f I (L), ( x] It fllws that ( x] H = ( x] Thrfr x ( x], ad h is a x = a x = a, si a x Thus x is miimal Nw, supps x L suh that x is miimal Si a Thrfr x = a x si x is miimal Thus x, si a Thus idal, w a hs tais prisly all miimal lmts i L Crllary 47: Lt L b a ASL with a miimal lmt Th, fr ay y x is miimal x, y L, x y is miimal if ad ly if Prf: W hav, fr ay idal I f L, ly if y x is miimal Alrady, w hav bsrvd that a mpty subst f a b a idal Nw, w hav th fllwig thrm x y I if ad ly if y x I It fllws that x y is miimal if ad ASL L whih is lsd udr th biary prati d t 06, IJA All Rights Rsrvd 67
G Naai Ra, Trf Gtahw By* / Idals i Almst Smilatti / IJA- 7(5), ay-06 Thrm 48: Lt I b a idal f L ad K b a mpty subst f L whih is lsd udr th prati with I K = Th thr xists a prim idal P f L suh that I P ad P K = Prf: Writ T = {J I (L) I J ad J K = } Th T, si I T Clarly T is a pst udr st ilusi ad it a b asily vrifid that T satisfis th hypthsis f Zr s lmma Thrfr, by Zr s lmma, T has maximal lmt say P W shall prv that P is prim Lt x, y L suh that x P ad y P Th P ( x] P ad P ( y] P Thrfr ( x] P, ( y] P T, si P is th maximal lmt i T H (( x ] P) K ad (( y ] P) K Chs t, t L suh that t (( x] P) K ad t (( y] P) K Th w hav t, t K ad h t t K Als, t ( x] P ad h t t ( x] P Similarly, t t ( y] P It fllws that t (( ] ) (( ] t x P y P) Nw, (( x] P) (( y] P) = (( x] ( y]) P = (( x] ( y]) P = ( x y] P If x y P, th ( x y] P H ( x y] P = P Thus t t P Thrfr t t P K whih is a traditi t P K = H x y P Thus P is a prim idal f L Thrfr thr xists a prim idal P f L suh that I P ad P K = Crllary 49: Lt I b a idal f L ad a I Th thr xists a prim idal P f L suh that I P ad a P Crllary 40: If 0 a L, th thr xist a prim idal P f L suh that 0 P ad a P Crllary 4: Lt I b a prpr idal f L Th th itrsti f all prim idals f L taiig I is I itslf Prf: Supps I is a prpr idal f L ad writ T = { P P is a prim idal f L ad I P} Put J = P W shall prv that I = J Si I P fr all P T, I J Cvrsly, supps J I Th P T x J suh that x I By Crllary 49, thr xists a prim idal P f L suh that I P ad x P Thrfr x J whih is a traditi t x J Thus J I H I = J thr xist Crllary 4: Lt a, b L ad ( b] Th thr xists a prim idal P f L taiig a ad t taiig b r vi vrsa Prf: Supps a, b L suh that ( b] Th ithr ( b] r ( b] Withut lss f grality, assum that ( b] Th a (b] Thrfr { a } ( b] = Nw, by rllary 48, thr xists a prim idal P f L suh that ( b] P ad { a } P = Thus b P ad a P Similarly, w a prv that a P ad b P Crllary 43: If a L is t miimal, th thr xists a prim idal f L t taiig a Prf: Supps a is t a miimal lmt f L Th thr xist x L suh that x a a Thus ( x a] Supps ( x a] Th a = ( x a] H a = ( x a) a = x ( a a) = x a whih is a traditi t a x a Thus by Crllary 4, thr xist a prim idal f L t taiig a Thrm 44 Th fllwig ar quivalt, i L Th itrsti f all prim idals f L is mpty Th itrsti f all prim idals f L is agai a idal 3 L has a miimal lmt I th fllwig thrm w a s that, if a idal I f L tais a uimaximal lmt, th a idal I ad a ASL L ar qual 06, IJA All Rights Rsrvd 68
G Naai Ra, Trf Gtahw By* / Idals i Almst Smilatti / IJA- 7(5), ay-06 Thrm 45: Lt L b a ASL ad m L b uimaximal If I is a idal i L suh that m I, th I = L Prf: Supps I is a idal f L ad supps m I is a uimaximal W shall prv that I = L Nw, lt x L Th x = m x Thrfr x I H L I But, I L Thus I = L Fially, w haratriz amiabl substs i L r prisly, w prv that if L has amiabl st, th vry amiabl st is ismrphi t th smilatti PI (L) First w d th fllwig Lmma 46L: Lt b a maximal st i L Th A (L), th st f all -amiabl lmts f L is a idal f L Prf: Supps is a maximal st i L Th larly A (L) is mpty, si vry lmt i is -amiabl Lt a A (L) ad t L Th thr xists m suh that m a= a Nw, a t = ( m a) t = m ( a t) Thus a t A (L) Thrfr A (L) is a idal f L Rall that if is a maximal st ad prprty x x = x ad x x = x x L is -amiabl, th thr xists a uiqu lmt i with th Nw, w hav th fllwig whs prf is straight frward Lmma 47: Lt b a amiabl st i L Th fr ay x L, ( x] = ( x ] x Thrm 48: Lt b a amiabl st i L Th fr ay ( x ] = ( y] x y ( ] = ( ] 3 x = y x, y L, th fllwig ar quivalt: Prf: Supps is a amiabl st i L Th A = L Nw, lt x, y L = A Th thr xists a uiqu lmt x, y suh that x x = x ad x x = x ad als y y = y ad y y = y Thus by rllary 34 ad lmma 36, w gt ( x ] = ( x ] ad ( y ] = ( y ] Thrfr () () is lar Assum () Th x ( x ] = ( y ] Thus x = y x = x y, si x, y H x y Similarly w gt y x Thrfr x = y Assum (3) W d t shw that ( x ] = ( y] Lt t (x] Th t = x t = ( x x) t = ( x x ) t = x t = y t = ( y y ) t = ( y y) t = y t ( y] H (x] (y] Similarly w a prv that ( y] ( x] Thrfr ( x ] = ( y] Crllary 49: Lt b a maximal st i L Th fr ay x = y ( x ] = ( y] x, y, th fllwig ar quivalt Rall that if is a maximal st i L ad fllwig thrm a, th fr ay L x, x a Fially w prv th Thrm 40: Lt b a amiabl st i L Th th mappig x (x] is a ismrphism f t a smilatti PI (L) Prf: Lt b a amiabl st i L Dfi, f : PI (L) by f ( x) = ( x] fr all x Th f is bth wll dfid ad - Als, lt (x] I (L) Th x L = A Thrfr thr xists a suh that a x = x Si x a, w gt f ( x a) = ( x a] = ( a x] = ( x] H f is t Nw, it rmais t shw that f is a hmmrphism Supps x, y Th f ( x y) = ( x y] = ( x] ( y] = f ( x) f ( y) 06, IJA All Rights Rsrvd 69
G Naai Ra, Trf Gtahw By* / Idals i Almst Smilatti / IJA- 7(5), ay-06 Thrfr f is a hmmrphism Thus f is a ismrphism Frm th abv thrm, vry amiabl st is mbddd i th smilatti PI (L) REFERENCES addaa Swamy, U ad Ra, G C: Almst Distributiv Latti, JAustral aths(sris 3(98),77-9 ari Ptrih, Psyvaia,: O Idals f a Smilatti, Czhslvak athmatial Jural, (97)97, Praha 3 Naai Ra, G ad Trf Gtahw By: Almst Smilatti, Itratial Jural f athmatial Arhiv- 7(3), 06, 5-67 4 Szasz, G: Itrduti t Latti Thry, Aadmi prss, Nw Yrk ad Ld, 963 Sur f supprt: Nil, Cflit f itrst: N Dlard [Cpy right 06 This is a Op Ass artil distributd udr th trms f th Itratial Jural f athmatial Arhiv (IJA), whih prmits urstritd us, distributi, ad rprduti i ay mdium, prvidd th rigial wrk is prprly itd] 06, IJA All Rights Rsrvd 70