A Note on Generalization of Semi Clean Rings

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1 teratioal Joral of Algebra Vol. 5 o A Note o Geeralizatio of Semi Clea Rigs Abhay Kmar Sigh ad B.. Padeya Deartmet of Alied athematics stitte of Techology Baaras Hid Uiversity Varaasi-5 dia Abstract this aer we have itrodced the otio of semiclea ideals -semiclea rigs ad -semiclea rigs which are a atral geeralizatio of semi clea rigs. t has bee show that matrix ideal over a semiclea ideal is semiclea. A sfficiet coditio for a semiclea ideal to be exchage ideal has bee rovided. t has bee show that matrix rig (R) over -semiclea ad -semiclea rig is -semiclea rig ad -semiclea rig resectively. Semiclea rigs are -semiclea. athematics Sbject Classificatio: 6D7 6P7 6E5 Keywords: Semiclea rigs semiclea ideals -Semiclea rigs -Semiclea rigs Gro rigs Exchage rigs. itbh8@gmail.com : trodctio: Let R be a ital rig. We say that R is a semiclea rig i case every elemet of R is the sm of a eriodic elemet ad a it elemet i R. t is well kow that every edomorhism rig of a cotably geerated vector sace over a divisio rig is clea hece semiclea. A rig R is said to it reglar i case for every x R there exist a it R sch that x = xx. Camillo ad Y [] claimed that every it reglar rig is clea hece semiclea. this aer a atral roblem is asked whether there is a o semiclea rig R while some elemet of R is the sm of a eriodic elemet ad a it elemet. So as to

2 4 Abhay Kmar Sigh ad B.. Padeya deal sch rigs we will itrodce a otio of semiclea ideal. We show that semiclea ideal have similar roerties to semiclea rigs. We call a rig R is -semiclea if every elemet of R ca be writte as sm of its ad a eriodic elemet ad which is a atral geeralizatio of semiclea rig. We will rove that if G is a cyclic gro of order 3 the gro rig Z G is -clea bt ot - semiclea for ay rime. A elemet x i R is -semiclea if there exists a ositive iteger sch that x is a -semiclea i R. A rig R is -semclea if every elemet of R is the sm of a eriodic elemet ad a fiite mber of its i R. All -semiclea rigs are -semiclea. A examle shows that -semiclea rigs are roer geeralizatio of -semiclea rigs for a ositive iteger.. Semiclea deal Defiitio.: A ideal of a ital rig R is semiclea i case every elemet i is the sm of a eriodic ad a it elemet i R. Clearly every ideal of a semiclea rig is semiclea. Bt there exist a o semiclea rig which cotais some semiclea ideal. Set R = R R where R is semiclea ad R is ot semiclea. The R is ot semiclea. Set = R give ay ( x) we have a eriodic elemet R ad it R sch that x = + becase R is semiclea. Hece ( x) = ( ) + (-) clearly ( ) is a eriodic elemet ad (-) is ivertible. Therefore we coclde that is a semiclea ideal of R. Hece the otio of semiclea ideal is o-trivial geeralizatio of semiclea rigs. Theorem.: [Theorem 5. []]. Let R be torsio free rig ad x R x = + where is eriodic is it. f = ± the R is clea. Lemma.: [Lemma. []]. Every clea ideal of a ital rig is a exchage ideal. Lemma.3: f its of a rig are ± the every semiclea ideal of a torsio free ital rig is a exchage ideal. Proof: The roof is clear from Theorem. ad Lemma.. Theorem.4: Let R be torsio free ital rig ad be a ideal i which every eriodic elemet is cetral. The followig coditios are eqivalet: (i) (ii) is a semiclea ideal. is a exchage ideal. Proof: ( i) (ii). Clear from Lemma. ( ii) (i). Sice i case of ital rigs i which idemotet is cetral every ideal is clea. Hece is a semiclea ideal.

3 A ote o geeralizatio of semi clea rigs 4 Theorem.5: Let be a semiclea ideal of a ital rig R. The matrix over is a semiclea ideal of (R). Proof: Clearly the reslt holds for =. Assme ow that reslts hold for = k (k ). Sose that A = k write a y A = Where a B k x B Sice is a semiclea ideal of R we have = R for some ositive itegers m ad ( m ) ad U(R) sch that x = + Sice B x y k the there exist P m = P k for ositive itegers m ad ( m ) ad Q GL (R) k sch that B x y = P + Q Set P = P = y U x Q + x y. t is easy to check that there exist a ositive itegers m ad sch that m P = P where m ) ( m ad + Q yq x x Q yq = + Q yq x x U Q yq U = k k Hece U GL (R) clearly A = P + U k roof. Therefore k is semiclea ideal of (R). By idctio we comlete the Corollary.6: Let R be a torsio free ital rig i which its are ± ad is a exchage ideal i which every eriodic is cetral. The is a semiclea ideal of (R). Proof: The roof is clear by Theorem.4 ad.5.

4 4 Abhay Kmar Sigh ad B.. Padeya The morita cotexts deoted by ( A B N ψ φ) cosists of two rigs A B ad two bimodle A N B A B ad a air of bimodle homomorhism (called air rigs) ψ : N B A ad φ : A N Bwhich satisfy the followig associative coditio: ψ ( m) = φ(m )m φ (m )m = mψ ( m ) for ay mm. N. This coditio isres that the set T of geeralized a matrices a A b B N ad m forms a rig called the rig of m b cotext. H. Che ad. Che stdied the clea morita cotexts with zero air rigs. Now we ivestigate the semiclea morita cotexts with zero air rigs. Lemma.7: Let T be the rig of morita cotexts ( AB N ψ φ) with zero air rigs. N f ad J are semiclea ideals of A ad B resectively the is a semiclea J ideal of T. Proof: Let T be the rig of morita cotext ( A B N ψ φ) with zero air rigs it is N easy to check that is a ideal of T. J a N Let A = where a b J m ad N. As is a m b J k l semiclea ideal of A the there exists a eriodic elemet A ad = for ay ositive itegers k ad l where ( k l) U(A) sch that a = +. as mch b J there is a eriodic elemet J ad v U(B) sch that b = + v. Set P = U = it is easy to check that there exist ositive itegers s ad t m b s t ( s t) sch P = P ad U v m v v = v m v v U = T Hece U U(T) obviosly we have A = P + U therefore we get the reslt.

5 A ote o geeralizatio of semi clea rigs 43 Let X Y ad Z be associative rig with idetities ad ad be A A) (A 3 A ) ad ( A3 A) bimodles resectively. Let φ : y ( be a ( ZX) homomorhism ad let oeratios. Theorem.8: The followig coditios are eqivalet: X T = Y be with sal matrix Z () J ad K are semiclea ideals of X Y ad Z resectively () The formal triaglar matrix ideal J is a semiclea ideal of K X Y. Z X Proof: ( ) (). Clearly J is a ideal of Y. Let K Z Y B = ad =. Sice J ad K are semiclea ideal of Y ad Z Z J resectively by lemma.7 we see that is a semiclea ideal of B. Agai by K X X Lemma.7 Y is a semiclea ideal of as reqired. Z B X () (). J is a ideal of Y we sow that J ad K are K Z ideals of X Y ad Z resectively. For ay a J we have

6 44 Abhay Kmar Sigh ad B.. Padeya K J a. Ths we have eriodic elemet T 3 ad a it T 3 sch that + = 3 3 a Clearly m = for ay ositive itegers m ad ) m ( ad ). U(Y Frther more we have. a + = Ths J is a semiclea ideal of Y. Likewise we claim that ad K are semiclea ideal of X ad Z resectively. Theorem.9: Let R be a ital rig ad a ideal of R. The followig coditios are eqivalet: () is a semiclea ideal of R. () Triaglar matrix ideal R R R L O L L is a semiclea ideal of the rig of all lower triaglar matrices over R. Proof: The roof is immediate coseqece of Theorem Semiclea Rigs ad -Semiclea Rigs: Defiitio 3.: Let be a ositive iteger. A elemet x of a rig is called -semiclea if x = K where is a eriodic elemet ad K are its i R. A rig is called a -semiclea rig if every elemet of R is -semiclea. Now we give some basic roerties of - semiclea rigs. Proositio 3.: Let be ositive iteger. The followig coditios hold: () A homomorhic image of a - semiclea rig is - semiclea. () A direct rodct = R R of rigs ( ) R is - semiclea rig if ad oly if each R is semiclea. Proof: Proof of first art is clear. We eed oly to rove ()

7 A ote o geeralizatio of semi clea rigs 45 Sose each R is a semiclea rig. Let for each of write x = + + K + where i U(R )( i ) ad is a eriodic elemet of i ( R ). The x = K+ where i = ( ) U( R )( i ) ad = ( ) P( R ) where P(R ) is a set of eriodic elemet of R. So R is a - semiclea rig. The coverse is immediately follows from (). The followig reslt is for the rig R[(x)] of all formal ower series over R. Proositio 3.3: The rig R[(x)] is - semiclea if ad oly if R is - semiclea. Proof: Sose f = a + ax + a x + K i R[(x)]. f a = K + i R where m is eriodic ( = m are ositive itegers) ad K are its i R the f = + ( + ax + a x + K ) + + K+ where is eriodic elemet of R the is also a eriodic elemet of R[(x)] ad + ax + a x + K U(R[(x)] i U(R) U(R[(x)] ( i ). Ths R[(x)] is a - semiclea rig. The coverse also holds becase R is a homomorhic image of R[(x)] which is a - semiclea rig. t is demostrated by Xiao ad Tog [9] i 5 that for ay ideal of R which is cotaied i J(R) for ay ositive iteger R/ is - clea ad idemotet ca be lifted modlo the R is - clea. Defiitio 3.4: Let is a ideal of a rig R. We say that eriodic ca be lifted modlo k l k l if for ay a R with a a. there exists b R sch that b = b R ad a b. Proositio 3.5: Let be a ositive iteger ad be a ideal of R which is cotaied i J(R). f R/ is - semiclea ad eriodic ca be lifted modlo the R is - semiclea rig. Proof: The roof follows from [Proositio.6 [9]] t has bee roved by Xiao ad Tog [9] i 5 that if e is a idemotet elemet i R sch that ere ad ( e)r( e) are both clea rigs the R is -clea for ay ositive iteger. We ca exted this reslt to - semiclea rigs. Proositio 3.6: Let be a ositive iteger ad e be a idemotet elemet i R. f ere ad ( e)r( e) are both semiclea rigs the R is also -semiclea. Proof: The roof is followed from [Theorem. [9]].

8 46 Abhay Kmar Sigh ad B.. Padeya Corollary 3.7: Let be a ositive iteger. f = e + e + Kem i a rig R where the e i are orthogoal idemotet elemets i R ad each e Re i i is - semiclea the R is also - semiclea. Proof: By Proositio 3.6 ad idctio. Corollary 3.8: Let be a ositive iteger. f R is a - semiclea rig the so is matrix rig m (R) for a ositive iteger m. Corollary 3.9: Let be a ositive iteger. f = K m are modles ad Ed ( i) is - semiclea for each i the Ed() is - semiclea. By Proositio 3.7 ad Proositio 3.6 we obtai Corollary 3.: Let be a ositive iteger. f A ad B are rigs ad = B A is a A bimodle the formal triaglar matrix rig T = is - semiclea if ad oly if B both A ad B are - semiclea. By above corollary ad idctive argmet the followig fact is clear. Corollary 3.: Let be a ositive iteger. The for each iteger m a rig R is - semiclea if ad oly if so is the rig T of all m m lower (res. er) triaglar matrix over R. Defiitio 3.: A elemet x of R is said to be - semiclea rig if there exists a ositive iteger sch that x is - semiclea. A rig R is called a -semiclea if every elemet of R is a sm of eriodic ad fiite mber of its i R. Clearly semiclea rigs are - semiclea. Let R = Z be rig of itegers. The R is - semiclea rig bt it is either semiclea or a - semiclea rig. Hece the class of a semiclea rig is a roer sbset of class of - semiclea rig. Proositio 3.3: Let be a ositive iteger. The followig coditios hold: () Homomorhic image of a - semiclea rig is - semiclea. () A direct rodct R = R of rigs ( R ) is - semiclea rig if ad oly if each R is semiclea.

9 A ote o geeralizatio of semi clea rigs 47 Refereces [] V.P. Camillo H.P. Y Exchage rig its ad idemotets Comm. Algebra (994) [] H. Che. Che O clea ideals JS 6() [3] J. Ha W. K. Nicolso Extesio of clea rigs Comm. Algebra 9() [4] Herike Two class of rigs geerated by their its J. Algebra 3(974):8-93. [5] W.K. Nicolso Liftig idemotet ad Exchage rigs Tras. Amer. ath. Soc. 9(977) [6] W.K. Nicholso Strogly clea rigs ad fittig lemma Comm. Algebra 7(999) [7] P. Ara Extesios of exchage rigs J. Algebra 97(997) o [8] W.K. Nicolso K. Varadaraja Cotable liear trasformatios are clea Proc. Amer. ath. Soc. 6(998) o [9] Xiao Gagshi Tog Wetig -clea rigs ad weakly it stable rage rigs Comm. Algebra 33(5) 5-57 [] Y. Yaqig Semiclea Rigs Comm. Algebra 3() (3) Received: Aril 5 8