Left Quasi- ArtinianModules

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Aerica Joural of Matheatic ad Statitic 03, 3(): 6-3 DO: 0.593/j.aj.03030.04 Left Quai- ArtiiaModule Falih A. M. Aldoray *, Oaia M. M. Alhekiti Departet of Matheatic, U Al-Qura Uiverity, Makkah,P.O.Box 5699, Saudi Arabia Abtract thi paper we tudy a ew cla of left quai-artiia odule. we how: if i a left quai-artiia rig ad M i a left -odule, the (a) Soc(M) e M ad (b) ad(m) all i M.The we prove: if i a o-ilpotet left ideal i a left quai-artiia rig, the cotai a o-zero idepotet eleet. Fially we how that a coutative rig i quai-artiia if ad oly if i a direct u of a Artiia rig with idetity ad a ilpotet rig. Keyword Module with Chai Coditio, Left Quai-Artiia Module ad ilpotet ig. troductio By rig we ea a aociative rig that eed ot have a idetity. thi paper, we tudy a ew cla of left quai-artiia Module, which i a geeralizatio of left Artiia odule. Firt we tudy the proble of fidig coditio which are equivalet to the defiitio of left quai-artiia Module(Theore.). The we how that the cla of left quai-artiia Module i Q-cloed, S-cloed ad E-cloed. ectio two we tudy the odule tructure over left quai-artiia rig, i particular we prove that if i a left quai-artiia rig, the every fiitely geerated left -odule M i a left quai-artiia(theore.)fially we how that: f be a rig, = (), the i a left quai-artiia if ad oly if i ilpotet ad each of the,, 3, i left quai-artiia -odule (Theore.4). ectio three we decribe the ideal tructure ad we give oe claificatio, i particular we prove that if i a o-ilpotet left ideal i a left quai-artiia rig, the cotai a o-zero idepotet eleet (Theore 3.). ext we prove that if i a ei-prie left quai-artiia rig ad be a o-zero left ideal of, the =e for oe o-zero idepotet e i (Theore 3.5)... Defiitio ad Baic Propertie Let M be a left -odule. We ay that M i a left quai-artiia Module if for every decedig chai...... of left -ubodule of M, there exit Z + uch that for all. t i c lear that ay left Art iia odu le i left * Correpodig author: fadoary@uqu.edu.a (Falih A. M. Aldoray) Publihed olie at http://joural.apub.org/aj Copyright 03 Scietific & Acadeic Publihig. All ight eerved quai-artiia ad it i eay to prove the followig Lea. Let M be a left -odule. (a) f M= 0, the M i a left quai-artiia. (b)f ha a idetity ad M i uitary,the M i left quai-artiia if ad oly if M i left Artiia. ow we prove the followig which i a characterizatio of left quai-artiia odule. Theore. Let M be a left -odule. The the followig coditio are equivalet: ς of left -ubodule of M uch (a) every o-epty collectio K ς, the K ς,there exit a iial eleet. that if (b) For every decedig chai of left -ubodule...... there exit Z + uch that a decedig chai teriate. (c)m i left quai-artiia. (d) For every o-epty collectio ς of left -ubodule of M, there exit ς ad Z + uch that K for ayk ς, K. (a (b) Suppoe that...... i a decedig chai of left -ubodule of M but the decedig chai...... of left -ubodule of M doe ot teriate for all Z +. Therefore the collectio ς {,,...,,,...,, i a oepty collectio of -ubodule ad for all ς,...} we have ς. Hece ha o iial eleet, which

Aerica Joural of Matheatic ad Statitic 03, 3(): 6-3 7 i a cotradictio. (b) (c) Let...... be ay decedig chai of left -ubodule of M the there exit Z + uch that...... for a decedig chai of left -ubodule of M ad by (b) there exit Z + uch that for all, but for all. Take t = ax {, } t the t for all, hece M i a left quai-artiia. (c) (d) Let be a o-epty collectio of left -ubodule of M uch that for each ad Z +, there exit K uch that K K. ow let ς ς uch that,where hece there exit, but the there exit, but 3, uch that 3, where 3 cotiuig i thi aer we ca cotruct a ifiite decedig chai...... of left -ubodule of M uch that,=,,.hece for oe, which i a cotradictio. (d) (a) Let be a o-epty collectio of left K for all K ς. K ς, for all Z +. But K K for -ubodule of M uch that The all Z +, hece by (d) there exit a Z + uch that K K for all Z +.Therefore if, the K K ad ha a iial eleet. ext we prove the followig: Propoitio.3 Let M be a left -odule. f M i left Artiia, the M i left quai-artiia. be a decedig chai of left -ubodule of M, Let... -uboduleof M. the... i a decedig chai of But M i left Artiia, hece there exit Z + uch that =. Therefore. For all Hece M i left quai-artiia. eark: The covere of Propoitio.3,eed ot be true a the followig exaple how: Let Q M Q quai-artiia 0 0 ad 0 Q 0. The M i left 0 -odule,but 0 0 M i ot left Artiia. Q 0 ow let Т be a cla of odule. The we ay that Т i S-cloed if i a ubodule of M ad M Т, the Т.We ay that Т i Q-cloed if M Т ad i a ubodule of M, the M Т. We ay that Т i E-cloed if i a ubodule of M ad, M Т, the M Т. Propoitio.4 Let Т be the cla of left quai-artiia odule. The (a)т i S-cloed. (b) Т i Q-cloed.(c) Т i E-cloed. (a) i clear (b) Suppoe that M i a left quai-artiia -odule ad i ubodule of M. Let π: M M = M be the atural hooorphi of left quai-artiia odule oto M. The i a decedig chai of ubodule of M, ad i a decedig chai of - ubodule - of M, where i π ( ) but M i left i quai-artiia, hece there exit Z + uch that for all. But k = k. Hece for all. Therefore Mi left quai-artiia. (c)suppoe that be a -ubodule of M ad, M Т. Let be a decedig chai of left -ubodule of M. The...... i a decedig chai of -ubodule of. But Z + uch that left quai-artiia, hece there exit i ( ) for all. ow + + i a decedig chai of ubodule ofm adm i left quai-artiia, therefore there exit k Z + uch that k ( k + ) + for all. That i k k + + for all. ow let = ax {, k} The ( ) ad + + for all. ow = + [ + ] ad by odular law, = + ( ) for all. ( ) Therefore ( ( ( ) ) ) for all Hece for all. Therefore M i left quai-artiia. A iediate coequece of Propotio.4, we have the followig Corollary Let Т be the cla of quai-artiia odule.f M = A+B

8 Falih A. M. Aldoray et al.: Left Quai- ArtiiaModule where A,B i Т the M Т. M where eark: Suppoe that ha,o M= M { : M} adm { : M}. Here M i uitary ad left quai-artiia if ad oly if M i left So M. M 0 quai-artiia if ad oly if M i left Artiia.Ad M i left are Artiia. Artiia if ad oly if M ad M. The Subodule Structure M thi ectio we tudy the ubodule tructure by coider odule over left quai-artiia rig. Firt we prove the followig Theore. Let be a left quai-artiia rig. The every fiitely geerated left -odule i left quai-artiia Let M be a fiitely geerated left -odule, the M = x + x + + x where 0 x i M, i. f = the M i cyclic ad therefore ioorphic to L where L = a ax = 0. Sice i left quai-artiia, o i every factor odule. Aue iductively that the Theore hold for odule which ca be geerated by - or fewer eleet. Thex i left quai-artiia ad M x x + x + x x (x + x ) x (x + + x ) which i left quai-artiia. Therefore M i left quai-artiia. Let be a rig ad M i a left odule. The (a)soc M = K M K i iple i M = L M L i eetial i M (b) ad M = K K i axial ubodule i M = L L i all ubodule i M Theore. Let be a left quai-artiia rig ad M i a left -odule.the (a) ocm e M (b)admallim (a) Let 0 x M. The ρ x : x uch that ρ x r = rx(r ) i a hooorphi of oto the ubodule x with Kerel Kerρ x = l x = r rx = 0. So l (x) x. But i left quai-artiia, hece by Propoitio.4, x i left quai-artiia. We clai that x cotai a i ial ubodule. To p rove thi let l = x 0 x M, M be a oepty collectio of -ubodule of x ad J l The J = y for oe 0 y M.But J = y = y = y y = J l.but l ha a iial eleet, hece Soc() 0.But Soc x = x Soc(M), hece Soc M em. (b) Firt we how that ad M = JM where J = J().Sice for ay left -odule M the factor odule ad(m ad(m)) = 0. Therefore M ad(m)i ubdirect product of iple left - odule. But icej()i aihilate all iple left -odule, o it aihilatem ad(m) that i JM ad(m). Coverely ice J i ei-iple the we have Soc M = r M (J) Therefore Soc M JM = r M JM J J = r M JM 0 = M JM. Hece M JM i ei-iple J-odule. Sice J i cotaied i aihilator of every iple -ubodule of M, the M JM i ei-iple -odule, thu ad M JM = 0 but ad M ad M = 0. Therefore ad(m) JM. Hece ad M = JM. ow ice left quai-artiia, aue J = 0 for oe Z + ad coider a -ubodule K of M withjm + K = M. Multiplyig with J we obtai J M + JK = JM, the J M + JK + K = M. Cotiue i thi way we have after tep,k = J M + K = M. Hece JM all i M therefore by firt part,ad(m) all i M. Corollary.3 Let be left quai-artiia rig ad M left -odule, the M i fiitely geerated if ad oly if M ad(m) i fiitely geerated. By Theore., ice ad(m) all i M, the the reult follow. By the il radical =() of a rig we ea the u of all ilpotet ideal of, which i a il ideal. t i well kow [7. P.8 Theore ], that i the u of all ilpotet left ideal of ad it i the u of all ilpotet right ideal of. ow we give aother characterizatio of left quai-artiia rig,aely the followig: Theore.4 Let be a rig, = () be the il radical of, the i a left quai- i ilpotet ad each of,, 3, Artiia if ad oly if i left quai-artiia -odule. Suppoe i left quai-artiia. The by[3,corollary.3] i ilpotet. ow let M. The M i left i quai-artiia -odule ad i a ideal of for all i. i Therefore i a -ubodule of M for all i.but by Propoitio.4, i i left quai-artiia for all i. Alo i i+ i -ubodule of i+ o each i i+ i left quai-artiia. To prove the covere, ote that ice it follow fro Propoitio.4, that i left quai-artiia -odule ad by iductio i i left quai-artiia for all i. But i ilpotet, hece there exit Z + uch that = 0, therefore i left quai-artiia -odule.

Aerica Joural of Matheatic ad Statitic 03, 3(): 6-3 9 Hece i left quai-artiia rig. 3. The deal Structure thi ectio we tudy the ideal tructure i a left quai-artiiarig. ote that if = Q 0, the 0 0 0 i a ilpotet ideal of. There ad Q 0 Q are left quai-artiia, 0 but i ot left quai-artiia. Hece the cla of left quai-artiia rig i ot E-cloed, however we have the followig: Theore3. A fiite direct u of left quai-artiia rig i a left quai-artiia. By iductio, it i eough to prove the reult whe = are left quai-artiia. Let where,...... be a decedig chai of left ideal of.the ad...,but, ideal of there exit r, uch that ) r ( ) i a decedig chai of left ideal of i a decedig chai of left are left quai-artiia rig,hece r r ) ( ad (.Let =ax{r,}.the ) ( ) ( ) ( ad ( for all. But ),hece for all ad for all.therefore i left quai-artiia. Theore3. Let be a o-ilpotet left ideal i a left quai-artiia rig, the cotai a o-zero idepotet eleet. To prove thi we eed the followig lea. Lea3.3 Let be a left quai-artiia rig. The every o-ilpotet left ideal of cotai a iial o-ilpotet left ideal. Let be a o-ilpotet left ideal of ad uppoe that doe ot cotai a iial o-ilpotet left ideal of. The 0 ad i ot ilpotet. Therefore there exit a o-ilpotet left ideal 3. Hece 0 ad i ot ilpotet. thi way we ca fid a o-ilpotet left ideal the 0 ad i ot ilpotet ad o o. Hece...... i a ifiite decedig chai of left ideal of which i a cotradictio. Therefore cotai a iial o-ilpotet left ideal of. Proof of Theore Let be o-zero o-ilpotet left ideal of. Sice i a left quai-artiia rig, the by Lea3.3, cotai a iial o-ilpotet left ideal K. Sice K 0 the there exit. However xk K x K uch that xk 0 ad xk i a left ideal of, hece by iiilty of ek K we have xk =K. Therefore there exit uch that xe x ad ice xe xe we get thatx ( e e) 0. ow, let K o { ak xa 0 }, therefore Ko i a left ideal of ad K o K ice, xk 0, for all x K. Therefore we ut have Ko 0 ad ( e e ) Ko. Hece e e. Sice xe x 0 we have that e 0. ow, e K i a left ideal of ad cotai e e 0, o that e 0, the e e K. Hece e. Corollary3.4 f i left quai-artiia rig, the every il left ideal of i ilpotet. Let be a o-zero il left ideal of ad uppoe that i ot ilpotet. The by Theore 3., there exit a ozero idepotet eleet e ad e. Therefore e i ilpotet which i a cotradictio. Hece ut be ilpotet. ext we prove the followig Theore3.5 Let be a ei-prie left quai-artiia rig ad be a ozero left ideal of, the = e for oe ozero idepotet e i. Sice i ot ilpotet, it follow fro Theore 3.,that cotai a o-zero idepotet eleet ay, e. Let the the et of left ideal A( e) { x xe 0} L { A( e) 0 e e } i ot epty. ow, if A(e) L, the A(e)L. ow ice i a left ideal of, the re, where r, e, therefore 0 re re, but i a left quai-artiia, hece by Theore.,Lha a iial eleet A ( e 0 ), ay. Either A( e 0 ) 0 or A ( e 0 ) = 0. f A ( e 0 ) 0, the A ( e 0 ) ut have a idepotet e, ay. By defiitio of A ( e 0 ), e ad e e 0 0. Coider e e0 e e0e, the

30 Falih A. M. Aldoray et al.: Left Quai- ArtiiaModule e ad i itelf a o-zero idepotet eleet. Moreover, e e e ( e0 e e0e ) e 0, hece e 0. ow if A(e ), the xe 0 ad x e 0 + e e 0 e = 0. Therefore x e 0 + e e 0 e e 0 = 0 ad xe 0 = 0. Therefore x A( e 0 ) ad A( e) A( e0), ice e A( e 0 ) ad e A( e ) we have that A( e) A( e0), which cotradict the iiality of A ( e 0 ). Therefore A( e 0 ) =0. But ( x xe0 ) e0 0 for all x hece ( x xe0 ) A( e0) 0 ad x xe for all x, 0 which ip lie that e e0. Hece 0 e 0. Corollary3.6 Ay ei-prie left quai-artiia rig i a ei-iple left Artiia. By Theore 3.5 every o-zero left ideal of i geerated by a o-zero idepotet e, ay. But we kow that e act a right idetity for the left ideal =e, ad ice i itelf a ideal, hece ha a idetity eleet. Therefore i left Artiia. ow, J() i ilpotet, ad i a ei-prie rig, iplie that J() = 0. Hece i a ei-iple. ow we decribe left quai-artiia rig uig the o coutative verio of Wedderbur Theore. particular we prove the followig Theore3.7 A coutative rig i quai-artiia if ad oly if i a direct u of a Artiia rig with idetity ad a ilpotet rig. To prove thi we eed the followig Lea3.8 Let be a left quai-artiia rig ad be the il radical of. The i a ei-iple Artiia rig. Sice i ilpotet ad follow that i left quai-artiia, it i a ei-prie left quai-artiia. Therefore by Corollary 3.5, i a ei-iple Artiia rig. Proof of theore3.7 Suppoe that i a direct u of a Artiia rig with idetity ad a ilpotet rig, ice ay Artiia rig ad ay ilpotet rig are quai-artiia,it follow that i a quai Artiia rig. To prove the covere. Let = () be a il radical of. The by Corollary 3.4, i ilpotet ad by Lea 3.8, i a ei-iple Artiia rig. Therefore by Wedderbur' Theore i a fiite direct u of it iial ideal, each of which i a iple Artiia rig, that i..., where i e i i a iial ideal of which i a iple Artiia rig. But a fiite direct u of Artiia i agai Art iia, hece i i a Artiia rig i ad i a ei-iple Artiia. But i i a ei-iple Artiia o, it ha a idetity eleet. Therefore i i a Artiia rig with idetity. Hece, i i ad i a direct u of Artiia rig i with idetity ad ilpotet rig. Fially we prove the followig which characterize the prie ideal i left Quai-Artiia rig. Theore3.8 Let be a coutative quai-artiia rig ad be a iial ideal i. The a( ) i a axial ideal. To prove thi we eed the followig Lea3.9 f i a coutative quai-artiia rig,the every prie ideal of i axial. Let P be a prie ideal of, the P i a prie rig. ow P i a ei-prie quai-artiia rig. Therefore by Corollary 3.5 P i a ei-iple Artiia.Hece by Wedderbur' Theore P i a fiite direct u of iial ideal, each of which i a iple Artiia rig. But a prie rig caot be writte a a direct u of o-trivial ideal, hece P i a iple rig. Therefore P i axial. A iediate coequece of Lea 3.9 we have the followig Corollary3.0 f i a quai-artiia rig, the J()= rad()= (). Where J() i the Jacobo radical of ad rad( ) ithe prie radical of. Proof of Theore3.8 By Lea 3.0, it eough to how that a( ) i a prie ideal i. Let x, y uch that x, y a( ). The x 0 ad y 0, but x ad y. But x ad y. 0 xy ad xy 0. Hece a( ) i a iial ideal of, hece Therefore xy, ad a( ) i a prie ideal of.

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