Nilpotent Elements in Skew Polynomial Rings

Joural of Scece, Ilac epublc of Ira 8(): 59-74 (07) Uvery of Tehra, ISSN 06-04 hp://cece.u.ac.r Nlpoe Elee Sew Polyoal g M. Az ad A. Mouav * Depare of Pure Maheac, Faculy of Maheacal Scece, Tarba Modare Uvery, P.O. Bo 45-34, Tehra, Ilac epublc of Ira eceved: 4 Deceber 05 / eved: 3 May 06 / Acceped: 5 February 06 Abrac Le be a rg wh a edoorph ad a -dervao. Aoe uded he rucure of he e of lpoe elee Aredarz rg ad roduced l- Aredarz rg. I h paper we roduce ad vegae he oo of l- (,) - copable rg. The cla of l- (,) -copable rg are eeded hrough varou rg eeo ad ay clae of l- (,) -copable rg are coruced. We alo prove ha, f l- -copable ad l-aredarz rg of power ere ype wh l lpoe, he l( [[ ; ]])()[[ l; ]]. We how ha, f a l- Aredarz rg of power ere ype, wh rg, he l ;, l lpoe ad l- (,) -copable l ;,. A a coequece, everal ow reul are ufed ad eeded o he ore geeral eg. Alo eaple are provded o llurae our reul. Keyword: (,) copable rg; Sew polyoal rg; Sew power ere rg. Iroduco Throughou h arcle, all rg are aocave wh dey. Le be a rg, be a edoorph ad a -dervao of, ha a addve ap uch ha ab ab a b, for all a, b. We deoe ;, he Ore eeo whoe elee are he polyoal over, he addo defed a uual ad he ulplcao ubec o he relao a a a for ay a. We alo deoe he ew power ere rg [[ ; ]], whoe elee are he power ere over, he addo defed a uual ad he ulplcao ubec o he relao a a for ay a. ecall ha a rg reduced f ha o ozero lpoe elee. Aoher geeralzao of a reduced rg a Aredarz rg. A rg ad o be Aredarz f he produc of wo polyoal zero ple ha he produc of her coeffce are zero. Th defo wa coed by ege ad Chhawchhara [6] recogo of Aredarz proof [4, Lea ] ha reduced rg afy h codo. Accordg o Aoe [3], a rg called lf g l ple Aredarz, f ab l (), for all * Correpodg auhor: Tel: +988883446; Fa: +988883493; Eal: ouav.a@gal.co 59

Vol. 8 No. WINTE 07 M. Az ad A. Mouav. J. Sc. I.. Ira. 0 0 f () a,() g b [] Whe a -pral rg, he he polyoal rg [ ] ad he Laure polyoal rg [, ] are -pral ad l-aredarz, ad l ( [ ]) l ()[ ]. Th codo rogly coeced o he queo of wheher or o a polyoal rg [ ] over a l rg l, whch relaed o a queo of Aur []. Th rue for ay -pral rg (.e. he lower l radcal Nl () * cocde wh l () ). I [3], M. Habb ad A. Mouav, ay, a rg wh a edoorph α l-aredarz of ew f. g l( ) [[ ; ]] power ere ype, f a b l, for all, ad for ple ha all 0 0 f () a,() g b [[ ; ]]. I h paper, we are cocered wh l-aredarz rg of ew power ere ype, whch a geeralzao of l-aredarz rg. Accordg o Krepa [5], a edoorph of a rg called rgd f a () a 0 ple a 0 for each a. A rg called -rgd f here e a rgd edoorph of. I [9], E. Hahe ad A. Mouav, ay a rg -copable f for each a, b, ab 0 f ad oly f a b 0. Moreover, ad o be - copable f for each a, b, ab 0 ple a b 0. If boh copable ad - copable, ad o be (,) -copable. By [], called wea copable, f ab l f ad oly f a b l for each a, b, ad ad o be wea copable f for each a, b, ab l ple a b l. Ufyg ad eedg he above oo, we ay a l- -copable rg f for each a, b, ab l f ad oly f a b l. Moreover, we ay l- -copable f for each a, b, ab l a b l. ple If boh l- -copable ad l- - copable, we ay ha l- (,) -copable. We eed he cla of l-(,) -copable rg hrough varou rg eeo. We how ha a l-(,) -copable rg f ad oly f he rg of ragular ar T l- (,) -copable, where a -dervao of T. If a l- Aredarz rg of power ere ype ad l-(,) - copable he ; a l- (,) -copable rg, where a -dervao of ;. A a coequece, everal propere of (,) - copable rg are geeralzed o a ore geeral eg. We how ha f a l--copable ad ll Aredarz rg of power ere ype wh lpoe, he l l [[ ; ]] [[ ; ]]. We alo how ha, f l-aredarz rg of power ere ype ad l-(,) -copable, wh l lpoe, he l ;, l ;,. Moreover we how ha, whe l-(,) - copable, -pral, ad eher a rgh Golde rg or ha he acedg cha codo (a.c.c.) o deal or ha he a.c.c. o rgh ad lef ahlaor or a rg wh rgh Krull deo, he ( ;, ) l ;,. l eul ad Dcuo We fr roduce he cocep of a l-(,) - copable rg ad udy propere. Defo.. For a edoorph ad a - dervao, we ay ha l- -copable f for each a, b, ab l f ad oly f a b l. Moreover, ad o be l- - copable f for each a, b, ab l ple a b l. If boh l- - copable ad l- -copable, we ay ha l- (,) -copable. By [9], -rgd rg are (,) -copable. Clearly every (,) -copable rg ad hece every -rgd 60

Nlpoe Elee Sew Polyoal g rg alo l-(,) -copable. Alhough he e of (,) -copable rg arrow, we how ha l- (,) -copable rg are ubquou. By [], a rg l-aredarz of power ere f. g l a b ype f [[ ]] l, for all, ad f () a, () g b ple [[ ]]. 0 0 Lea.. Le be a l-(,) -copable rg. The () ab l () f ad oly f a ()() b l, for each pove eger uber. () ab l() ple a ()(), b l for each pove eger uber. (3) If a l-aredarz of power ere ype ad a b l () he ()()(), a b l p q ()()() a b l whe,, p, q are pove eger uber. Proof. () Sce l-(,) -copable, we have he followg plcao: ab l ()()() a b l a () b a ()() b l. Coverely we have a () b l () a (())() b l a () b l () ab l (). () Th lar o (). (3) ab()l ple ba()l becaue for u ba ple ha u bra, for each r, u ()()() bra bra br ab ra. Bu ab l() he ab l () ce l- Aredarz of power ere ype, hu o u l () ad ba l (). We have u l () ab l (), o a ()() b l by (). The we have ()() b a l o ()()() b a l ad ha ()() a b l (). The we coclude ha ()() a b coaed l (). We do he ae for p q ()() b a l () wh pove eger p, q. Lea.3. Each wea (,) copable rg l-(,) -copable. Proof. Suppoe ha ab l (). So arb l () for each r. The ()() ar b l ad o we have ()() ar b l (), by wea (,) copably. So a ()() b l. Slarly, ()() ar b l for all r ad by wea (,) copably we have ()()() ar b l ad o a ()() b l. Ne aue, a ()() b l. The ()()() ar b l ad by wea (,) - copably, we have ()() ar b l for all r, o ab l (). I he followg, we wll ee ha he covere o rue. Ideed, here e a rg, whch l-(,) -copable bu o wea (,) -copable. Thu a l-(,) -copable rg a rue geeralzao of a wea (,) -copable rg (ad hece (,) - copable rg). We he ca fd varou clae of l- (,) -copable rg whch are o wea (,) -copable ad hece are o (,) -copable. Eaple.4. Le K be a feld, ad S K, y, z. Le S K, y, z. y Alo aue ha a edoorph of S ad be a edoorph of, gve by: (),(),() z. z f (),(),() z. z We fr how ha well defed. To ee h, le g for oe f, g, o here e h, f, f S uch ha f g f yf. ()() f g ()()()() f y f Thu ()()() g f yz f. So ()() f g, ad well defed. Now, we deere he e of lpoe elee of. Fr, we fd zero devor ooal. Le 0 f be a zero devor ooal. 6

Vol. 8 No. WINTE 07 M. Az ad A. Mouav. J. Sc. I.. Ira We have f uf u,g v gv, wh u, u, v, v, y, z, f, g uch ha fg 0, o uf u. v gv 0. If u v y oe of uf or y, he here e f or f u or v g or g or gv. Bu f oe of hee cae occur, f 0 or g 0. u v So y, ad u y, v. Hece 0 f lef (rgh) zero dvor f ad oly f f f, g g y. If f a lpoe ooal of, he f f y. Moreover () f y0. So f a lpoe ooal, f ad oly f f f y, for oe ooal f. Now we cla ha, f f a lpoe polyoal ad f f where f ooal for oe, he f lpoe (.e. f f y polyoal f for oe ). Before provg he cla we have he followg propery: The deg(f) z, where f a ooal he fro r r y z y z y z r r, Slarly, deg(f), deg(f) y are defed by r r repecvely. Alo deg()() f. r. ad Proof of he cla: A f deg(f) aal ad f o lpoe. Le Aue ha f, f, f A are he ooal uch ha deg, deg y, degz hey have he large. Oe ca ee ha a lea oe of deg,deg y,degz ozero. Whou lo of geeraly, le deg 0. Sce f o lpoe f o zero devor, hece () f o zero. Alo, worh o ay ha he ooal wh large deg f () f. So ca o be plfed ad h ea ha f o lpoe. Th coradco how ha deg() f deg() deg() f 0 f. So f y z lpoe ad coa whch ea ha f 0. Hece f eher zero or he for f y for oe f. Now, le fg l (). Suppoe ha f uf, g gv ad u, v, y, z. If u, v y, he fzg l (). So we have f f, g g y, hece f ()()(). g f g y l I obvou ha f ()() g l. Coverely le f ()() g l, wh f uf, g gv ad u, v, y, z. Sce f ()() g l, u,() v v y, o v y. Hece fg f g y, whch obvouly a ube of l (), whch how ha a l- -copable rg. Bu eay o ee ha y z l (), whle y () z yz 0(). l Thu o wea - copable. Noe ha l () o a deal of. Th becaue y z l (), z, bu z y z l (), y z z l (). Le be a -dervao of. The edoorph of eeded o he : edoorph T ()() T defed by (())(()) a a, alo he -dervao eeded o he -dervao T ()() T : defed by (())(()) a a, for each a T (). The we have he followg. () Theore.5. A rg l-(,) -copable f ad oly f he ragular rg T () l- (,) - copable. Proof. Suppoe ha a l-(,) -copable A a B b T(). rg ad (),() We how ha AT ()(()) B l T AT ()()(()) B l T. We oberve ha l (()) T l () 0() l. 0 0 0() l f ad oly f The for C (r) T (), we have AC(()) B l T 6

Nlpoe Elee Sew Polyoal g ar b 0(()) ar b l T 0 0 ar b a r b l() for a r ()() b l, for, by l- (,) copably - ar () b 0()(()) ar b l T 0 0() ar b AC() B l (()) T AT ()()(()) B l T. The cae l- -copably lar. Ne uppoe ha T() a l-(,) -copable rg ad ha a, b, r, A () a,() B, b C(c) are dagoal arce T (). The we have a b l () arb 0 0 0 arb 0 0 l(()) T 0 0 arb AC(()) B l T AC()(()) B l T, for all r, by l- (,) -copably ar () b 0 0 0() 0 ar0 b l(()) T 0 0() ar b ar ()() b l for all r a ()() b l. The cae l- -copable lar. Le be a rg ad le a a a 0 a a S () ad, a a 0 0 a a a a 0 a a wh 0 0 a, ad le T (, ) be he rval eeo of by. Ay edoorph of ca be eeded o a edoorph of S () or T (, ) or T (,) a T (, ) defed by (())(()) a a, ad ay - dervao ca be eeded o a -dervao of S() (or T(,) or T(,) ) defed by (())(()). a a Theore.6. Le be a edoorph ad a -dervao of. The he followg codo are equvale: () l-(,) -copable. () S () l- (,) -copable. (3) T(,) l- (,) -copable. (4) T (,) l- (,) -copable. Proof. Ug he ae ehod a he proof of Theore.5, he reul follow. By [, Lea.9], proved ha, f - l l. A edoorph pral he ad -dervao of a rg [ ], gve by : [ ] [ ] are eeded o defed by ( a )= a, ad [ ] 0 0 defed by ( a )= a 0 0 : [ ]. We ca ealy ee ha a -dervao of he polyoal rg [ ]. Lea.7 Le be a l-aredarz rg of power ere ype, l- -copable ad a ()() b l, a, b, uch 63

Vol. 8 No. WINTE 07 M. Az ad A. Mouav. J. Sc. I.. Ira ha 0,,,,. The a ()() b l for all. Proof. We have he followg ye of equao: a b l (); 0 0 0 a b a ()(); b l 0 0 a b a ()()()(). b a b (*) a b l 0 0 We wll how ha a ()() b l by duco o. If 0, he a 0 b 0 l (), () b 0 a 0 l. Now, uppoe ha a pove eger uch ha a ()() b l, whe. We wll how, whe. Mulplyg ha a ()() b l equao (*) by b0 fro lef, we have b a ()()()() b b b a b b a b b a b b a b 0 0 0 0 0 0 0 0 0 By he duco hypohe a ()() b0 l, for each,0. So a ()() b0 l by [, Lea 3], hece a b0 l (), by lcopably. The a b0 l (), b0 a l (), for each,0. Thu b0a ()() b0 l ad o b a ()() b l, o b0a b0 l (), ad 0 0 hece a ()() b0 l. Mulplyg equao (*) by b, b,, b fro he lef de repecvely, yeld a ()(),()(), b l,() a b l a b l, 0 ur. Th ea ha a ()() b l, whe. Theore.8. If a -pral ad l-(,) - copable rg, he he polyoal rg [ ] l- (,) -copable. Proof. Le f g l f () a, 0 p 0 r r [ ], we have 0, wh g ( ) b [ ]. The for all f r g l. Hece p () arb l = 0 a r b () l for 0,,,, p. Bu -pral, o a rb l (), by ehod of Lea.7, ad by l-(,) -copably we have a r () b l () for all,,. Thu a r() b l (). So we ca coclude ha p f r ( g ) (() 0 a r b l f () g l how ha f g l. Hece we ge. Slarly, we ca. The covere lar. Thu [ ] a l-(,) - copable rg. Le be a -dervao of, ad for eger, wh 0 f Ed,, wll deoe, he ap whch he u of all poble word, bul wh leer ad leer. For ace f, f, f ad 0 0 0 f appear [6].. The e lea Lea.9. For ay pove eger ad r we have ab a, b r f () r he rg ;,. 0 By [7], a rg l-ecouave f l ple ab l., for all Lea.0. Le be a l-, -copable rg ad l-aredarz of power ere ype. If a b l af () b l for all 0. he Proof. Ug Lea., he proof rval. By [, Lea 3], f a l-aredarz rg of power ere ype he a l-ecouave rg. 64

Nlpoe Elee Sew Polyoal g Now we have: Propoo.. Le be a l-(,) -copable rg ad l-aredarz of power ere ype. The we l ( ;,() ) ; l,. have Proof. Le f () a ( [) ;, ]. 0 l There e a pove eger uch ha 0. 0. So l. Thu we have... f The we have a a a a lower er a α a α a α a 0 aα a α a α a α a l, by [, Lea 3]. So we have a α a α a α a α a l, by l-(,) -copably. I ple ha a α a α a α a.. α a l, 3 he aα a α a α a, by [, Lea 3], o 3... α a a l a α a α a α a a a l By followg h ehod, we have a l. Alo a. aò l, he a l have f a l f l for 0. 0 a. We, by Lea.0. So Now we f A a a a f A a A wh, - 0 -. The ;. Noe ha he coeffce of ca be wre a u of ooal a ad f ( a ), where a, a a, a,, a ad 0, are 0 oegave eger ad each ooal ha a f ( a ). Becaue l-aredarz of power ere ype, we ge ha l () ;, he er A a a a or. Now coder 0, o we have A l ; α, δ, he A a α a α a +lower er l ; α, δ () a α a α a l () (). Hece. A he argue above, he we have a l (). By followg h ehod we have a l 0. Hece f l ( ) [ ;, ]. for Corollary.. If 0, he l ( [ ; ]) l ;. Lea.3. Le be a l-(,) -copable ad l-aredarz rg of power ere ype rg. The l l a. Proof. Le u l o u a wh l (). Hece a a l(), ad by δ- copably we have a ()(). a l So a l (), he a l(). Theore.4. Le be a l-(,) -copable, l-aredarz rg of power ere ype ad l lpoe. The ;, Proof. Le f () l [ ]( [ l ]);,. a l ;, 0 [ ]. For a ay arbrary coeffce of f, δ a l. A α a edoorph, α a l. The here e aural uch ha ( l ) zero. Now we coder f v v ()...(). a f u a f u a All coeffce 0... of v v f are he for of ()...() u u whch he produc of eber of he a f a f a, l by Lea.0, o hould be equal o zero. Thu f l ; α, δ ). 0 f ad hece ( [ ] 65

Vol. 8 No. WINTE 07 M. Az ad A. Mouav. J. Sc. I.. Ira Corollary.5. Le be a l-(,) -copable, l-aredarz rg of power ere ype ad l lpoe. The we have l( [ ;, ]) l [ ;, ]. Corollary.6. Le be a l-(,) -copable ad -pral rg. Aue ha eher a rgh Golde rg or ha he acedg cha codo (a.c.c.) o deal or ha he a.c.c. o rgh ad lef ahlaor or a rg wh rgh Krull deo. l ( [ ;, ]) l [ ;, ]. The Proof. If ha ay of hee cha codo, he upper lradcal Nl () of lpoe. If ha he a.c.c. o deal, Nl () ca be characerzed a he aal lpoe deal of. If ha he a.c.c. o boh lef ad rgh ahlaor, Nl () lpoe by a reul of Here ad Sall [0, Theore.34], whle lpoe by a reul of f rgh Golde, Nl () La [8, Theore ]. Alo, f a rg wh rgh Krull deo, he by [7], Nl () lpoe. Corollary.7. If a l-aredarz rg of l lpoe ad l- - power ere ype ad copable rg, he l( [ ]) l ; ;. Proof. By Corollary.5, f 0 l( [ ; ]) l ;., he Theore.8. If a l-aredarz rg of power ere ype ad l-(,) -copable rg l lpoe, he ; copable rg. a l- (,) - Proof. Aue ha,), ad le f () a g( b ; 0 0 ; () ; p a ;, f g l. For all r 0 we have f r () g l.; Corollary.7, we have f r () g So by p l (()()) a r ; b l 0 The ;. a l ()() r b (), for 0,,,, p. Bu l-aredarz rg of power ere ype, o a ()()(), r b l by Lea.7, wh 0,0, 0 p. A lecouave, o we ge a ()() r l (), he a ()() r b l () Lea.. The ()() b, by a r b l () a r b l (). Hece ()() we have p (()()) 0. a r b So l ; l ;. Therefore we ge f r g l ; α. Coverely aue f ; α g l ; α. So we have p l ; f r g (()()) a r b, 0 for all r ; α. Thu we have a ()() r b l(). Sce l Aredarz rg of power ere ype, a ()() r b l (), o a ()() r b l (). Hece a ()() r b l (), a ()() r b l (). o The for all,,, we have l (). p a ()() r b (()()) a r ; b l, ad 0 hece f r ( g ) l ; α So, for 66

Nlpoe Elee Sew Polyoal g all r ; α f ; α ( g ) l ; α. Fally we have. For he cae of l- -copably, le f ; α g l ; α. The we have f r g p (()())()[ a ; ], r b l 0 for all r ; α. Hece a ()() r b l (). The a ()() r b l () for 0,0,0 p. Baed o he aupo we have ()() r b a l (), o a () r b l () ad ha a ()() r b l (). Hece a ()(()) r b l(), ad ha p 0 (()(())) a r ; b l. Th ple f r () g l,; α for all ; α ; ( ) l ; α r. Therefore we coclude ha f α g. Now we coder he lpoe elee ew polyoal rg whe a l-aredarz rg of power ere ype. Theore.9. Le be a l-aredarz rg of power ere ype ad l-, -copable rg. Le ; α, δ. 0 0 f () a,() g b f α δ g l α δ If ;, ;,, he a b l () for 0, 0. a rb Proof. Le 0, 0. a b for all r, We have r ; α, δ o ;, f rg f α δ g hece we have, () r() a 0 0 ()(()) a rb l( [ ]) 0 0 ()(()) b ;,, a rb ( )[ ;, ] 0 0 l Propoo.. Therefore a () 0 f rb l ( ) [ ;, ], wh. Pu 0, a f () rb,,,,, hece l(). equao: a ()(); rb l o by We have he followg a ()() rb a()(); a rb f rb l ()()()(); a rb a f rb a f rb l (())(). a f rb l The ce l-ecouave by [, Lea 3], applyg he ehod he proof of [, Theore.4], we oba a ()() rb l, he a b l(). Theore.0. Le be a l-(,) -copable ad I be a l deal of uch ha I I, I I. The a l-(,) - I copable rg. Proof. We have o prove a b l f ad oly f a() b l, for ay a, b, uch ha a a I, b b I. Fr aue ab l ad r. The a r ()() b a b, o ai r I α b I a () b. The 67

Vol. 8 No. WINTE 07 M. Az ad A. Mouav. J. Sc. I.. Ira arα b l. Bu I a () b l () arb I, I l-(,) hece arb l (), o I I l. A -copable, we have arα b l arα b l. I I l () The a () b l I o. The cae l- - copable lar. Coverely aue a () b l ad arb ab. The ()arb I a b. Uder he aupo (), hece ar () b l a r b l. A l-(,) -copable, we have arb l for all r, o we cocluded ha ab l () l I. I Defo.. [3] A rg ad o be, - ew l-aredarz f wheever 0 0 f () a,() g b ; α, δ afy. l,,, he a b l,, for ay,. f g, Lea.. If a l-(,) -copable ad l-aredarz rg of power ere ype, he a, -ew l-aredarz rg. Proof. Le 0 0 0 f () a,() g b ; α, δ ad f. g l,,. Therefore a f () b l ( ) [ ;, ], wh. So (())() a f b l, ce l-ecouave by [, Lea 3], applyg he ehod he proof of [, Theore.4], we oba a f ()() b l wh. The a, -ew l-aredarz rg. Propoo.3. Le be a l-(,) -copable ad l-aredarz rg of power ere ype, he for each depoe elee e e l ad e e u uch ha u l,. Proof. We have e e αe ee. By ag polyoal f e e, g e e, we ee ha f. g 0, whch ple ha f. g l ; α, δ l ;, by Propoo.. e e ee e l(). Now ae h e () α e, () ee. The we have h. 0 ad o e (), o we ge e e () l l. Now ae p e e α e ad q e e α e ; α, δ. The p. q eα e e eα e e α e l ; α, δ, ce e l () ad l-aredarz rg of power ere ype. Bu, -ew l- Aredarz by Lea., o e. e α e eα e α e l ( ) (). Now ae e e() α e, ee() αe ; α, δ. The we have. e () α e e e () α e e e() αe eαe. A e l(),. l ; α, δ. 68

Nlpoe Elee Sew Polyoal g Ad o a e. e ( α e ), -ew l-aredarz rg, hu () (). Now by () ad () we oba u e α e l. Hece α e e u e eα e l wh u l (). Theore.4. Le be a l-(,) -copable ad l-aredarz rg of power ere ype. The for each depoe elee e ad a, ea ae u wh u l(). Proof. Accordg o he Propoo.3, e e u wh u l () l., e () Now ae he polyoal f e ea e, g e ea e ; α, δ. Hece f. g ea e e ea e. ea e. l ad, e () O he oher had, u l () l-aredarz rg of power ere ype. So we have = ea e α e ea e e ea e e ;,. ea e u eu e e l α δ Slarly ea eea e l ; α, δ f g l ; α, δ ge e. ea e l ea ea e l h e e ae,. The, hece we, ad ha (). Le e eae, accordg o a earler ae we have eeae l. Hece ae eae l (). Ug (), () we have ea ae l, o ea a e u wh ul. Defo.5. For a edoorph ad a -dervao, a deal I ad o be l- (,) - copable provded ha: ) ab l a b l. For all a, b I. ) a b l a b l. all a, b I. For Theore.6. Le be a abela l-aredarz rg of power ere ype. The he followg aee are equvale : ) a l-(,) -copable rg. ) For each depoe e wh e e u,, e ad u l() ad e l () e are l-(,) -copable deal. Proof. rval. Le e be a l-(,) - copable deal ad ab l for each a, b o arb l, hece earb l. Thu () ea rebl. Bu e a l-(,) -copable deal, hece we have ha ea rα e α b ea r e u α b ea reα b ea ru α b l. Sce u) l (, we have ea ruα b o ea reα b ea rα bl l, (). Now, accordg o he above argue for e, we have e arα b l Wh () ad () we oba arα b l all r or aα b l ()., for. For he cae of l- -copable, we do a lar way. Coverely uppoe ha aα bl, he we ge arα b l, for each r. Bu (ea)rα (eb)= (ea) rα (e)α (b)= (ea) r (e+u) α (b)= (ea) r e α (b)+ ea rα b ea ruα b ad ha (ea) ruα (b) 69

Vol. 8 No. WINTE 07 M. Az ad A. Mouav. J. Sc. I.. Ira ea r α b,ea ruα b l. The ea rα eb l, ce e l- (,) -copable deal, hu we have ea r eb ea r b l (3). Slarly, we have ea r b l (4). Therefore (3),(4) ple arb l, for all r. Hece ab l. We coue o eed l- -copable codo ad, ; α. If o, f () a, a =,. a, we defe, for each eger uber Theore.7. If a -pral ad l- - copable rg, he, copable rg. have Proof. Le,,, a l- - [ ]. The we,. Hece for, f a uber, wh he eger we have f a, hece f g 0 o f g,., Now by Theore.8, a l- -copable rg. ecall ha a rg called of bouded de of lpoecy, f here e a pove uber uch ha 0 l., for each Lea.8. [, Lea ] If a l- Aredarz rg of power ere ype, he l( [[ ]]()[[ ) l ] ]. Theore.9. Le be a l-aredarz rg of power ere ype ad of bouded de. The l( [[ ]]()[[ ) l ] ]. Proof. By Lea.8 uffce o prove ha l()[[ ]]( [[ l ]] ). Sce l- Aredarz of power ere ype, l l ad of bouded de, a a rg, by [, Theore.5]. The [[ ]] a l rg of bouded de. Hece we ge l ()[[ ]]( [ ]] l [ ). Lea.30. [, Lea] Le be a l- Aredarz rg of power ere ype. Le f, f,, f [[ ]] f f f l()[[ ]] ad. The for all coeffce a of aa a l(), f. Theore.3. Le be a l-(,) -copable ad l-aredarz of power ere ype rg wh bouded de. The [[ ]] a l- (,) - copable rg. Proof. Le 0 0 f () a,() g b [[ ]] ad aue ha f [[ ]] g l( [[ ]] ). For each we have r() c [[ ]] 0 f r()( g [[ ]] l ). If u a arbrary elee of f [[ ]] g he u f r g r [[ ]],, for all. Uder he aupo we have f r g l( [[ ]]()[[ ) l ] ] ce, l-aredarz of power ere ype o l ad ce l-(,) - a cb copable a () b l c ad 70

Nlpoe Elee Sew Polyoal g ac () b l() for 0,,, Hece 0. (()) ac b l()[[ ]], o we have ()[[ ]]( ) f r g l l [[ ]]. Ad h ea we prove ha f rg f [[ ]] g f [[ ]]( () [[ g ] l ]). Coverely, f [[ ]]( g [[ l ]] ). If, he by he aupo f r g l ( [[ ]]) l ()[[ ]], ad ce l- Aredarz of power ere ype, we have () l. Hece a cb l a c b l for 0,,, a c b ad ha. So () a cb 0 l()[[ ]] l [ ]) Ad h ea ha ( [ ]. f [[ ]]( g [[ l ]] ). For he cae of l- -copable, we do a lar ehod. The [[ ]] a l- (,) -copable rg. Defo.3. A rg wh a edoorph ew l-aredarz of power ere ype, f wheever for all, 0 0 f () a,() g b [[ ; ]] f ().() g l()[[ ; ]], he a b l for all,. Propoo.33. Le be a l- -copable ad l-aredarz rg of power ere ype. The ew l-aredarz rg of power ere ype. Proof. Le, 0 0 f () a,() g b [[ ; ]] f ().() g l()[[ ; ]], hu (()) a b l()[[ ; ]], o 0 a ()() b l, hu a ()() b l, for all, by Lea.7. The ew l- Aredarz rg of power ere ype. Lea.34. Le be a l-(,) -copable ad ew l-aredarz rg of power ere ype. The l-ecouave. Proof. Le r ad ab l. The a()(()) r r r b l()[[ ; ]]. So ar b l ad hece a r b l. Lea.35. Le be a ew l-aredarz rg f f f of power ere ype ad aue ha l()[[ ; ]]. The ()()() a a a l()[[ ; ]], for all coeffce a of f. Proof. We wll how ha a ()()() a a a l () by 3 duco o. Suppoe ha a ()()() a a a l () 3. Sce () a 3 a l for, we have a ()()()() a a a (). Th becaue, f a l, a a()(()()) b b b b ()[[ l ; ]]. So ab l ()() ( ) ()[[. ; ]] l a a a, we have ad hece Theore.36. Le be a l-(,) -copable l-aredarz rg of power ere ype. The l( [[ ; ]]()[[ ) l ; ] ]. Proof. We how ha l( [[ ; ]]()[[ ) l ; ] ]. Le f () l( [[ ; ]]). The f 0 a 0 for oe 7

Vol. 8 No. WINTE 07 M. Az ad A. Mouav. J. Sc. I.. Ira pove eger. So we have (()()()). 0 f a a a a 3 0 If a a arbrary eber of coeffce of f, a a a l [[ ; he () ]] ( e). Hece we have Lea., ad Lea.34, we have a Thu a l, ad hece f l [[ ; ]]. aα a α a α a l. The by l. Theore.37. Le be a l-(,) -copable ad l-aredarz rg of power ere ype ad l be lpoe. The l()[[ ; ]]( [[ l ; ]] ). Proof. Le f () a l()[[ ; ]] 0 a l ad α () a l Sce l eger uch ha () l 0. The for all. lpoe, here e a pove produc of elee fro l ad ay zero. Now coder f (()()()) a a a a 0 l()[[ ; ]] 3, o... a ()...() a a... ()...() 0 a a a l, he. Hece ad ha f ()( [[ l ; ]] ). f 0 Corollary.38. Le be a l-(,) -copable l-aredarz rg of power ere ype ad l be lpoe. The l()[[ ; ]]( [[ l ; ]] ). Theore.39. Le be a l-(,) -copable, l-aredarz rg of power ere ype ad l () be lpoe. The [[ ; ]] a l- (,) - copable rg. Proof. Le 0 0 f () a,() g b [[ ; ]] ad f (). [[ ; ]].()( g [[ ; l ]] ). The for all r () r [[ ; ]] 0 f ().( r.()( ) g [ ; l ]] [ ). If u f [[ ; ]] () g elee, he u f r () g we have a arbrary, for all r( ) [[ ; ]]. f ().( r.()( )) g [[ ; l ]]. l()[[ ; ]] Uder he aupo we have Sce ew l-aredarz of power ere ype, a ()() r b l. Sce a ()() r b l a ()() r b l copable, Hece l-(,) - ad for 0. (())) a r b l()[[ ; ] ] 0 (. The we ge f ().( r.(() ) g ) l()[[ ; ]]( [[ l ; ]] ). Ad h ea ha f [[ ; ]] () g l() [[ ; ]]. Coverely, we u prove ha f (). [[ ; ]].(( g [[ ) ; ]] l ). If u f ().( r.()().[[ g ; ]].(), f g he uder he aupo f [[ ; ]] () g l( [[ ; ]]()[[ ) l ; ]], ad ce ew l-aredarz of power ere ype we have a ()() r b l. Hece ()() l a ()() r b l a r b ad ha for 0. 7

Nlpoe Elee Sew Polyoal g Hece (()()) 0 a r b l()[[ ; ]] 0 f (). [[ ; ]].(( g [[ ) ; l ]] ), for. Ad h ea ha. For he cae of l- -copably, we ca do a lar way. Thu [[ ; ]] a l-(,) -copable rg. Theore.40. Le be a l-(,) -copable ad l-aredarz rg of power ere ype. If f ().()()[[ g ; l ]], he f (). [[ ; ]].()()[[ g ; ]] l for all f, g [[ ; ]]. r() Proof. Le f ().()()[[ g ; l ]] c [[ ; ]], 0, ad for aue ha [[ ; ]] u f r g f g. The Bu ().() 0 ((c)()). u a b (()) a b l()[[, ; ] ] 0 f g ad ew l-aredarz of power ere ype, o a b l for all,. By Lea.34, l-ecouave, whch yeld a α b l By Lea.,.. aα b l Thu a (c)() b l ad hece a (c)() l, for all,,,, (()()) 0 l()[ [ ; ]]. a r b b whch yeld Therefore we have f (). [[ ; ]].() g l [[ ; ]]. Corollary.4. Le be a ew l-aredarz rg of power erewe ype, ad l-(,) - copable. The [[ ; ]] rg. a l-ecouave Proof. We prove ha, f ().()( [[ ; l ]] ), he for all, [[ ; ]] we ge (). [[ ; ]].()( [[ ; ]]). We have f g f g f g l l( [[ ; ]]) l()[[ ; ]]. The [[ ; ]] a l-ecouave rg by Lea.34. We rear ha, he above reul eable u o produce large clae of rg whch afy he l ( [ ;, ]) l [ ;, ]. codo eferece. Aur A., Algebra Over Ife Feld. Proc. Aer. Mah. Soc. 7: 35-48 (956).. Alhevaz A., ad Mouav A., O Mood g Over Nl-Aredarz g. Co. Algebra 4: - (04) 3. Aoe., Nlpoe Elee ad Aredarz g. J. Algebra 39: 38-340 (008). 4. Aredarz E.P., A Noe O Eeo of Baer ad p.p.- rg. J. Aural. Mah. Soc. 8: 470-473 (974). 5. Breeer G. F., K J. Y., ad Par J. K., gh Prary ad Nlary g ad Ideal. J. Algebra 378: 33-5 (03). 6. Callo V., Kwa T. K., Lee Y., Ideal- Syerc ad Sepre g. Co. Algebra 4: 4504-459 (03). 7. Che W., O Nl-ecouave g. Tha J.Mah. 9: 39-47 (0). 8. Habb M., Mouav A., Alhevaz A., The McCoy Codo o Ore Eeo, Co. Algebra. 4(): 4-4 (03). 9. Hahe E., Mouav A., Polyoal Eeo of Qua-Baer g. Aca Mah. Hugar. 07: 07-4 (005). 0. Her I.N., Sall L.W., Nl g Safyg Cera Cha Codo. Caad. J. Mah. 6: 77-776 (964).. Hze S., A Noe O Nl Power Serewe Aredarz g. ed. del Crc. Ma. Palero. 59: 87 99 (00).. Huh C., K C.O., K E.J., K H.K., Lee Y., Nl adcal of Power Sere g ad Nll Power Sere g. J. Korea Mah. Soc. 4: 003-05 (005). 3. Habb M., Mouav A., O Nl Sew Aredarz g. Aa-Eur. J. Mah. 5: -6 (0). 4. Kawar P., A. Leroy A., Maczu J., Idepoe g Eeo. J. Algebra. 389: 8-36 (03). 5. Krepa J., Soe Eaple of educed g. Algebra Colloq. 3: 89-300 (996). 6. La T.Y., Leroy A., Maczu J., Pree, Sepree ad Pre adcal of Ore Eeo. Co. Algebra. 5: 459-506 (997). 7. La T.Y. A Fr Coure Nocouave g. Sprger-Verlag, New Yor, 397 p. (99). 8. La C., Nl Subrg of Golde g are Nlpoe. Caad. J. Mah. : 904-907 (969). 9. Lezer E. S., Wag L., Golde a of Sew Power Sere g of Auoorphc Type. Co. Algebra 73

Vol. 8 No. WINTE 07 M. Az ad A. Mouav. J. Sc. I.. Ira 40(6): 9-97 (0). 0. Luqu O., Jgwag L., Nl-Aredarz g elave o a Mood. Arab. J. Mah. (): 8-90 (03).. Luqu O., Specal Wea Propere of Geeralzed Power Sere. J. Korea Mah. Soc. 4: 687-70 (0).. Luqu O., Jgwag L., O Wea, Copable g. Ieraoal Joural of Algebra. 5: 83 96 (0). 3. Madya A., Mouav A., Paya K., g Whch he Ahlaor of ad Ideal I Pure. Algebra Colloquu. : 948-968 (05). 4. Mazure., Nle P., Zebow M., The Upper Nlradcal ad Jacobo adcal of Segroup Graded g. J. Pure Appl. Algebra 9: 08-094 (05). 5. Paya K., Mouav A., Zero Dvor Graph of Sew Geeralzed Power Sere g. Cou. Korea Mah. Soc. 30: 363-377 (05). 6. ege M. B., Chhawchhara S., Aredarz g. Proc. Japa Acad. Ser. A Mah. Sc. 73: 4-7 (997). 7. Wag Y., e Y., -good g ad Ther Eeo. Bull. Korea Mah. Soc. 50: 7-73 (03). 8. Zhag W.., Sew Nl-Aredarz g. J. Mah. 34: 345 35 (04). 74