FULLY RIGHT PURE GROUP RINGS (Gelanggang Kumpulan Tulen Kanan Penuh)

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1 Joual of Qualty Measuemet ad Aalyss JQMA 3(), 07, 5-34 Jual Pegukua Kualt da Aalss FULLY IGHT PUE GOUP INGS (Gelaggag Kumpula Tule Kaa Peuh) MIKHLED ALSAAHEAD & MOHAMED KHEI AHMAD ABSTACT I ths pape, we study the pue ght deal gs, ad the the fully ght pue gs. The ecessay ad suffcet codtos fo a g to be fully ght pue g ae detemed. Also, we study some popetes of fully ght pue gs. Fally we study the fully ght pue goup gs. The ecessay ad suffcet codtos o a goup G ad a g A fo the goup g A[G] to be fully ght pue goup g ae detemed. Keywods: pue deal; fully pue g; goup gs; fully ght pue goup gs ABSTAK Dalam makalah, kam megkaj uggula kaa tule dalam gelaggag, da seteusya uggula tule kaa peuh. Syaat-syaat cukup da pelu utuk suatu gelaggag mejad gelaggag tule kaa peuh dtetuka. Kam juga megkaj sfat-sfat bag gelaggag tule kaa peuh. Akhya kam megkaj gelaggag kumpula tule kaa peuh. Syaatsyaat cukup da pelu bag suatu kumpula G da suatu gelaggag A bag gelaggag kumpula A[G] utuk mejad gelaggag kumpula tule kaa peuh dtetuka. Kata kuc: uggula tule; gelaggag tule peuh; gelaggag kumpula; gelaggag kumpula tule kaa peuh. Itoducto Goup gs ae vey teestg algebac stuctues, the cocept s elatvely old. The cocept became mpotat afte the atcle of Coel, 963, whch gave hghly stmulatg to the aea as well as the cluso of chapte o goup gs the book by Lambek, 966. Thee ae may studes that dscuss the ecessay ad suffcet codtos fo a g to have all ts deals the same class, that s, all ght deals ae pme (Tsutsu 996), all ght deals ae sem pme (Ahmad 008), o all ght deals ae weakly pme (Hao et al. 00). The am ths pape s to deteme the ecessay ad suffcet codtos o a goup G ad a g A fo the goup g AG [ ] to be fully ght pue goup g (all ts ght deals ae pue) ad to deteme the popetes of ths kd of goup gs.. Pelmaes Notato.: Thoughout ths pape the followg otato wll be adopted l. a.. g : Utay assocatve g. : Left ahlato ght deal g. zc g : Zeo commutatve g. z g : Zeo setve g. U : The set of all uts of a g.

2 Mkhled Alsaahead & Mohamed Khe Ahmad N : The set of all lpotet elemets of. Z : The set of all zeo dvsos of a g. SN : The set of all smple lpotet elemets of x x o : The set of all dempotet elemets of. I AC g : Almost commutatve g. A ( A ) : The ght ahlato of the set A. Al ( A ) : The left ahlato of the set A. { : }. AG [ ] : The goup g of a goup G ove a g A. ( G ) : The augmetato deal of AG [ ]. w G [H] : The ght deal of AG [ ] geeated by the set{ h : h H }. : The augmetato map of the goup g. Defto.. () A elemet x s sad to be a ght (left) stogly egula f thee s y such that x yx x x y. s sad to be ght (left) stogly egula g f evey elemet of s ght (left) stogly egula. () A g s sad to be π-egula g f fo all x thee s ad y such that x x y x. () A g s sad to be sem π- egula g f fo all x thee s ad y such that x x yx o x = xyx. emak.3. (Kayyaly 008) It s well kow that s ght stogly egula g f ad oly f s left stogly egula, so we have stogly egula g oly. Defto.4. () A g s sad to be almost commutatve g ( AC g ) f fo ay pope ght pme deal P of ad a P, thee s x such that ax C P. () A g s sad to be a left ahlato ght deal ( l. a.. g ) ff, A ( A ) s a ght deal of fo all subset A of. () A g s sad to be duo g ff, all ts deals ae two sded deals. (v) A g s sad to be zeo setve g ( z g ) ff, fo all a, b ab 0, mples that a b 0 fo all. (v) A g s sad to be sem commutatve g ff, a a, a. (v) A g s sad to be zeo commutatve g ( z g ) ff, fo all a, b that ab 0, mples that ba 0. Lemma.5. (Sbah 999) Fo ay g the followg hold: () If s a zc g, the s a z g. () s a z g ff, s a l. a.. () If s a z g, the I C. g. l such that such 6

3 Fully ght pue goup gs Lemma.6. (Wog 979) If s AC g completely pme (sem pme) deal., the evey pme (sem pme) deal s a Lemma.7. (Ahmad & Hjjaw 04) If s a utay g satsfes oe of the followg codtos, the I C. () If SN {0}. () If s a zeo commutatve g. () If s a duo g. (v) If s a sem commutatve g. emak.8. If s a egula g whch SN {0}, the s a AC g. Poof. Let P be a ght pme deal ad a P, sce s a egula ad a, the thee s b such that a a b a P. ab s dempotet fo (by Lemma.7) ad ab such that ab abab aba b ab. Let ab e, so ab C sce I C P fo f ab P, the aba P cotadcto. Thus b ( ) ( ) ab C P, theefoe s a AC g. Lemma.9. Let be a AC g. If s a fully sem pme ght g, the s a stogly egula g. Poof: Let a ad suppose a a P ; P s pme deal (sce a s a sem pme ght deal),. The thee s P such that a P sce s AC g, the thee s x such that a x C ad axp. So we have a x a x a x a x a a x x a x a. Ths mples a x a P (sce a s a sem pme ght deal) cotadcto. So a a. Theefoe thee s such that a a, thus s stogly egula g. Defto.0. A pope ght deal I of a g s sad to be pue ght deal f fo all a I, thee s bi such that a ab. Theoem.. (Al-Jaleel 987) Let I be a deal of a commutatve g, the I s a pue deal f ad oly f fo evey fte subset a, a,..., a of I, thee exsts b I such that ab a fo all,,...,. Lemma.. Let A be a ght deal of a g. If I s a pue ght deal of A, the I s a pue ght deal of. Poof: Let M I I, the t s clea that M s a ght deal of ad we emak. I I I I I. I I I. A I IA I I I ( I I. I fo I s pue ght deal A ). So I I I M Ths meas that, I s a ght deal of ad t s pue ght deal. 7

4 Mkhled Alsaahead & Mohamed Khe Ahmad Theoem.3. Let f : S be a g epmophsm. If I s a pue ght deal of, the f I s a pue ght deal of S. Poof: Let f : S be a g epmophsm. Let I be a pue ght deal of ad y f I, the thee s x I f x y. Sce x I ad I s pue, the thee s x I such that x x x. such that Cosequetly, f x. f x f x. So y. f x y. But f x f I y f x f I such that y y y. Hece f I s a pue ght deal of S., so thee s 3. Fully ght Pue g Defto 3.. A g s sad to be a fully ght pue g ff, evey pope ght deal I of s a pue ght deal. s fully left pue g ff, evey pope left deal of s a pue left deal. s fully pue g f t s fully ght ad fully left pue g. Theoem 3.. (Kayyaly 008) Let be ay g, the the followg ae equvalet: () s a fully ght pue g. () s a stogly egula g. () s a abela egula g. Lemma 3.3. Let be a g whch SN 0, the fo all x s a two sded deal. Poof: Let x we have A x ad y A x, the xy 0, so yx yxyx y xy x 0, so yx SN 0. Ths mples yx 0. Thus y Al x. Theefoe A x Al x. I the same way Al x A x, hece A x s a two sded deal. Lemma 3.4. (Wog 97) Let be a g whch SN 0 ad evey completely pme deal s a maxmal ght deal. The Z U. Theoem 3.5. The followg ae equvalet fo ay g : () s a fully ght pue g. () s a egula AC g. () s a fully ght dempotet AC g. (v) s a fully ght sem pme AC g. (v) (v) 0 0 SN ad evey pope completely pme deal s maxmal ght deal. SN ad s a sem egula g. 8

5 Fully ght pue goup gs Poof: Sce s a fully ght pue g, the t s stogly egula (Theoem 3.). a 0 but s stogly egula, so thee s Now, let a SN, the such that a a 0. 0, theefoe SN 0. Sce s stogly egula, the t s egula. We have s egula ad SN 0, thus s AC g (emak.8). Clea. v Let I, P ae two ght deals of such that I P. Sce I s dempotet ght deal, the I I P ths mples I P, thus P s sem pme ght deal ad s fully ght sem pme g. v v Sce s a fully ght sem pme AC g, the s a egula g (Lemma.9). Now Let P be ay completely pme deal, ad suppose P I whee I s a ght deal of. The thee s a I such that a P. Sce s a egula g, thee s b such that a a b a whch mples a b a 0 P, sce P s a completely pme deal ad a P we have a b P I, but a I mples a b I. Theefoe I whch mples I. Ths meas that P s a maxmal ght deal, ad we have SN 0 (Lemma.9). v v Sce SN 0, the fo all 0 x we have (Lemma 3.3), so / A x s aga a g whch 0 If y SN y 0 y 0 y A x 0 y A yx A yx 0 yxyx x l yx SN {0} 0 We have A x s a two sded deal SN fo : yxy 0 0 y x y yx 0 yx y A x A x y 0. x x A x s ot zeo dvso fo f y such that xy. 0. The 0 xy A x so xyx 0 whch mples xyxy 0. y 0,.e. xy SN.e., 0 l yx 0 xy 0 whch mples xy whch mples y A x, so y 0, the (Lemma 3.4) x A x, so thee s p A x thee s such that x whch mples such that x p mples, mples xx px x whch mples xx 0. x x mples,, xx px x x x xx so s sem π- egula g. v Let I be a pope ght deal ad x I, the thee s ad such that x xx, let, the x xx x xx x xx x x x xx xx xx x x x xx. x xx. x x x x x. x x x 0 9

6 So x get x Mkhled Alsaahead & Mohamed Khe Ahmad xx SN 0 whch mples x xx 0 (Lemma.7). So x x x fully ght pue g. SN. Thus x xx x x, ad sce 0 Theoem 3.6. The followg ae equvalet fo ay g : () s a fully ght pue g. () s a egula ad sem commutatve g. () s a egula z g. (v) s a egula zc g.. Cotug ths pocess to SN, the I C xy whee y I. Theefoe I s pue ght deal ad s Poof: Sce s a fully ght pue g, the t s abela egula g (Theoem 3.). Now, let x a, the x ab a ab whee a I C, so x a ab ab a ab a a a, so a a, by the same way a a. Thus a a fo all a of ad s a egula ad sem commutatve g. Let ab 0, the fo all we have a b ab. Sce s sem commutatve g whch mples a b 0 0. So s egula z g. v Let 0 ab, the we have b a b b a fo s a egula g, but s z g, so b I C (Lemma.5) whch mples b a b b a b b a b a b b ab b 0 0. Thus s a egula zc g. v Let I be ay ght deal of ad a I. Sce s egula ad a, the thee s b such that a ab a but s zc g, so ab I C (Lemma.7). Ths mples a ab a ab a e a ae, whee e ab I. So I s pue ght deal, theefoe s fully ght pue g. Coollay 3.7. Let be a ght ata sem pme ad AC g, the s fully ght pue g. Poof: Sce s sem pme ad ght ata, the s egula by (Lambek 966, poposto 3.5.). Sce s egula ad AC g, the t s fully ght pue g (Theoem 3.5). Coollay 3.8. A g s fully ght pue g f ad oly f s fully completely sem pme ght g ad AC g. Poof Sce s a fully ght pue g, the s fully sem pme ght g ad AC g (Theoem 3.5). Sce s AC g, the evey sem pme ght deal s completely sem pme ght deal (Lemma.6). So s fully completely sem pme ght g ad AC g. 30

7 Fully ght pue goup gs Sce s fully completely sem pme ght g, the s fully sem pme ght g, by assumpto s AC g, so s fully ght pue g (Theoem 3.5). Coollay 3.9. Let be a fully ght pue g, the evey pme deal of s a maxmal ght deal of. Poof: Let P be ay pme deal of. Sce s fully ght pue g, the s AC g (Theoem 3.5), so P s completely pme deal (Lemma.6). Theefoe P s a maxmal ght deal (Theoem 3.5 (5)). Coollay 3.0. If s a fully ght pue g, the s duo g. Poof: Let be a fully ght pue g, so s stogly egula (Theoem 3.). Theefoe s egula ad SN 0. Sce SN 0, the I C (Lemma.7). Now, () Let I be ay ght deal of ad a I, the a e fo some e I sce s egula g, thus a e e a I, whch mples that a I fo all a I. Theefoe I s a two sded deal. () Let I be a left deal of ad a I, the a e ; e I, so a e e a I, thus I s a two sded deal. Hece s duo g. 4. Fully ght Pue Goup g I ths secto, we wll fd the ecessay ad suffcet codtos o a goup G ad a g A fo the goup g AG [ ] to be fully ght pue goup g, ad we wll study some popetes of ths kd of goup gs. Theoem 4.. (Coell 963) A goup g AG [ ] s a egula g f ad oly f: () A s a egula g. () G s a locally fte goup, ad () The ode of ay fte subgoup of G s a ut A. Lemma 4.. Let I be ay ght deal of a g A ad G be ay goup. If I [G] s a pue ght deal AG [ ], the I s a pue ght deal A. Poof: Let a I, the a I[G]. But I [G] s a pue ght deal AG [ ]. So thee s,,..., such that sdes, the we get Sce bg, whee b I, a a a b g, aplay (the augmetato map) o both a a b. b I,,,..., ad I s ght deal A, the b I 3

8 Mkhled Alsaahead & Mohamed Khe Ahmad Let b b, so a ab whee b I. Thus I s a pue ght deal A. Theoem 4.3. If the goup g AG [ ] s a fully ght pue goup g, the A s a fully ght pue g. Poof: Let AG [ ] be a fully ght pue goup g ad I be ay ght deal of A, the I [G] s ght deal of AG [ ]. But AG [ ] s a fully ght pue goup g, so I [G] s a pue ght deal. Hece I s a pue ght deal (Lemma 4.), theefoe A s a fully ght pue g. Theoem 4.4. Let A be a commutatve g ad G be abela goup, f: () A s a egula g. () G s locally fte goup, ad () The ode of ay fte subgoup of G s a ut A. The the goup g AG [ ] s fully pue goup g. Poof: Sce A s commutatve ad G s abela, the AG [ ] s commutatve goup g. So the dempotet elemets ae cetal. Sce (), () ad () hold, the (Theoem 4.) AG [ ] s egula g. Thus we have AG [ ] s a abela egula g, so t s fully ght pue goup g (Theoem 3.). Theoem 4.5. Let A be a fully ght pue g. If: () G s a locally fte abela goup, ad () The ode of ay fte subgoup of G s a ut A. The the goup g AG [ ] s fully ght pue goup g. Poof: Sce A s fully ght pue g, the A s abela egula g (Theoem 3.).e., A s egula ad IA CA. Sce A s a egula g, ad () ad () hold, the AG [ ] s egula g (Theoem 4.). Now, let a g A[ G], but AG [ ] s egula, so thee s b h A[ G] such that,.e. j j j m a g a g. b h. a g. Ths mples m a g a g. b h. a g ths mples j j j m a g a g. b j h ja g j m a g. b a h g. But A j j j s egula g, thus fo all a thee s c such that a ac a ad ac I A C A. Theefoe m a g a g. b ( a c a ) h g j j j m a g. b ( a c ) a h g j j j 3

9 Fully ght pue goup gs m a g. ( a c ) b a h g, sce ac I A C A. j j j m a g. ( a g ) c b a h, sce G s abela. j j j m a g. a g. c b a h. j j j Let m j c b a h j j, the AG [ ] s fully ght pue goup g (Theoem 3.)., hece AG [ ] s stogly egula g, thus Theoem 4.6. Let AG [ ] be a egula g. The AG [ ] s fully ght pue g f ad f AG [ ] s a l. a.. g. Poof: ) AG [ ] s fully ght pue g, the AG [ ] s duo g (Coollay 3.0), so t s l. a... g. ) sce AG [ ] s l. a.. g the t s z g (Lemma.5), theefoe I A[ G] CA [ G] (Lemma.5). Sce AG [ ] s egula g by hypothess, the t s abela egula, so AG [ ] s fully ght pue g (Theoem 3.). Defto 4.7. A goup G s called a Dedekd goup f ad oly f all ts subgoups ae omal. Theoem 4.8. Let AG [ ] be a fully ght pue goup g, the G s a Dedekd goup. Poof: Sce AG [ ] s fully ght pue goup g, the t s duo g (Coollay 3.0). Now, Let H be ay subgoup of G. Sce AG [ ] s duo g, the w G [H] s a two sded deal, so H s omal subgoup of G by (Coell 963, Poposto ()), theefoe G s a Dedekd goup. Coollay 4.9. Let AG [ ] be a fully ght pue goup g ad G be a o tval goup, the AG [ ] has at least oe o tval cetal dempotet. Poof: Let G e, the thee exst h G; h e suppose H h Sce G s locally fte (Theoem 4.), the O( H) fo some but AG [ ] s a egula g (Theoem 3.), so s vetble A (Theoem 4.), theefoe e h 0 s a o tval dempotet of AG [ ] (fo f e the h h... h. e 0 mples = 0 (sce the goup G foms a base fo the A module [ ] AG ) whch s ot tue). Also e s cetal sce [ ] AG s a abela egula (Theoem 3.). 33

10 Mkhled Alsaahead & Mohamed Khe Ahmad Poposto 4.0. If the goup g AG [ ] s a fully ght pue goup g, the the followg hold: () A s a stogly egula g. () A s a egula AC g. () A s a fully dempotet g. (v) A s a fully sem pme g. (v) A s a sem π- egula g. Poof: Dectly fom Theoem 4.3, Theoem 3. ad Theoem 3.5. efeeces Ahmad M.K. & Hjjaw S. 04. Cetal dempotet of gs. Joda Joual of Math. ad Stat. 7(): Ahmad M.K Fully sem pme ght gs ad egula popetes. J. of Aleppo Uv. 6: -. Al- Jaleel Y.O Pue deal commutatve gs wth uty. Maste Thess. Uvesty of Joda, Joda. Coell I.G O the goup g. Cod. J Math 5: Hao Y., Poo E. & Tsutsu H. 00. O g whch evey deal s weakly pme. Bull Koea Math. Soc. 47(5): Kayyaly I S-utal sem goup g ad pue deals those gs. Maste Thess. Aleppo Uvesty, Sya. Lambek J Lectue gs ad Modules. Tooto: Blasdell. Sbah A.D Some popetes of elemets goup gs. Maste Thess. Al al-bayt Uvesty, Joda. Tsutsu H Fully pme g II. Com. Algeba 4(9): Wog E.T Almost commutatve gs ad the tegal extesos. Math. J. of Okayama Uv. 8(): 05-. Wog E.T. 97. egula gs ad tegal exteso of a egula g. Ameca Math. Socety 33(): Depatmet of Mathematcs Al al-bayt Uvesty Mafaq, JODAN E-mal: mekhledsaheed@yahoo.com* Depatmet of Mathematcs Aleppo Uvesty Aleppo, SYIA E-mal: mk_ahmad49@yahoo.com * Coespodg autho 34