Symmetric Skew Reverse n-derivations on Prime Rings and Semiprime rings

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1 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) - Volume 47 Number July 7 Symmetric Skew Reverse -Derivatios o Prime Rigs ad Semiprime rigs B. Satyaarayaa, Mohammad Masta Departmet of Mathematics Acharya Nagarjua Uiversity Nagarjua Nagar 5 5 A.P., Idia. drbs6@yahoo.co.i chamasta9@gmail.com Abstract: Let be ay fixed positive iteger ad ä deote the trace of symmetric skew reverse -derivatio Ä:R R, associated with a atiautomorphism á.let I be ay Ideal of R.()If R is o commutative prime rig such that [ä(x),á (x)]=, for all x I the Ä = i R.()Let R be o commutative semiprime rig such that ä is commutig o I ad [ä(x),á(x)] Z(R), for all x I the [ ä(x), á (x)]= for all x I. Key words : Prime rig, semiprime rig, commutig mappig, cetralizig mappig, derivatio, skew derivatio, reverse derivatio, skew reverse derivatio, automorphism, atiautomorphism.. INTRODUCTION The cocept of reverse derivatios of prime rigs was itroduced by Bresar ad Vukma []. Relatios betwee derivatios ad reverse derivatios with examples were give by Samma ad Alyamai [7]. Recetly there has bee a great deal of work doe by may authors o commutativity ad cetralizig mappigs o prime rigs ad semi prime rigs i coectio with derivatios, skew derivatios, reverse derivatios, skew reverse derivatios [,4-6,7-,5-7,9-]. Vukma [9-], Mohammad Ashraf [], Jug ad Park [7] have studied the cocepts of symmetric biderivatios, -derivatios, 4-derivatios ad derivatios. Ajda Foser [5], Faiza Shujat ad Abuzaid Asari [7], Jayasubba Reddy et.al [8] ad Basudeb Dhara ad Faiza Shujat [4], Yadav ad Sharma [] have exteded ad studied the cocepts of symmetric skew -derivatios, 4derivatios ad -derivatios. Recetly Jayasubba Reddy et.al, [9-] have studied the cocepts of symmetric skew reverse derivatios. Ispired by these works we have made a attempt to itroduce the otio of symmetric skew reverse - derivatios ad prove some properties, which may be a cotributio to the theory of commutig ad cetralizig mappigs of prime rigs ad semi prime rigs.. Prelimiaries Defiitio... Let be ay fixed positive iteger ad a mappig : R R is said to be symmetric (permutig), if the equatio holds (x, x, x,..., x ) (x (), x (),..., x () ) xi R ad for every permutatio ë S, the Symmetric permutatio group of order. Defiitio... Let be ay fixed positive iteger ad a mappig :R R is defied as (x)= (x,x,x,.,x),where Ä:R R, is called the trace of the symmetric mappig. Defiitio.. A additive mappig Ä: R R is said to be symmetric skew derivatio, associated with a automorphism, if for every i =,,..., ad for all x, x,...x i, x i,...x R, the x (x, x,..., xi, x, xi,..., x ) is mappig a skew derivatio of R associated with the automorphism. I Throughout this paper R will represet a rig with ceter ISSN: -57 Z (R). Let be ay iteger. A rig R is said to be - torsio free, if for all x, y R, x = x =. Recall that a rig R is said to be prime, if a R b =, a, b R either a= (or) b= ad a rig R is said to be semiprime, if a R a =, a R a=. Let us deote the commutator xy - yx by [ x, y ].I this paper we extesively make use of the commutator idetities [ xy,z ]=[ x, z ] y + x [ y, z ] ad [ x, yz ] = [ x, y] z + y [ x,z]. Let I be ay o-empty two sided ideal of R. The the mappig : R R is said to be commutig o I, if [ (x), x ] =, x I ad cetralizig o I if, [ (x), x ] Z(R), x I. A additive mappig D:R R is called a derivatio if, D(xy)=D(x) y + x D(y) holds x, y R ad is called a reverse derivatio, if D(xy)=D(y) x + y D(x) holds x, y R. A additive mappig D:R R is called a skew derivatio ( -derivatio ) associated with a automorphism if, D(xy)=D(x) y + (x) D(y) holds x, y R ad is called symmetric skew reverse derivatio ( - derivatio) associated with a atiautomorphism if, D(xy)=D(y) x + (y) D(x), holds x, y R particular, for all x, y, x, x,..., x i, x i,..., x R, Page 8

2 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) - Volume 47 Number July 7 (x,x,...,xi,xy,xi,...,x) = (x, x,..., x i, x,x i,..., x ) y (x) (x, x,..., x i, y,x i,..., x ). Defiitio..4. A additive mappig Ä:R R is said to be symmetric skew reverse derivatio, associated with a atiautomorphism á, if for every i=,,..., ad for all x, x,..., x i, x i,..., x R, the mappig x (x, x,..., xi, x,xi,..., x ) is a skew reverse derivatio of R associated with the atiautomorphism á. I particular, for all x, y, x, x,..., xi, xi,..., x R, (x, x,..., xi, xy,x i,..., x ) = For coveiece let us defie (x y) as follows: (x y) = cr f r (x,y). r (x) cr fr (x,y) (y), x, y R, where r fr (x,y) (x,x,x,...,-r times, y,y,y,...,r times),! Cr. r!( r)! Example.5. Let F be a field ad á be a atiautomorphism of F x Suppose that R = (x, x,..., x i, y,x i,..., x ) x + (y) (x, x,..., x i, x,x i,..., x ) Further, (, x, x,..., x ) = (+, x,..., x ) (, x, x,..., x ) (, x, x,..., x ). Hece y : x, y F. Clearly, the set R of all x matrices w.r.t matrix additio ad matrix multiplicatio is a commutative rig. Let us defie a map x y ( x ) á : R R such that, x,y F. Obviously is a atiautomorphism. xk yk R, k=,,,...,. (, x, x,..., x ) Next let us deote A k (x x, x, x,..., x ) (x, x,..., x ) (-x, x,..., x ) Now let us defie a mappig Ä: R R such that ( x ) ( x )... ( x ) (A, A,..., A ) (-x, x,..., x ) (x, x,..., x ). Clearly, Ä is a odd fuctio if is odd ad is a eve fuctio, if is eve. If Ä: R R is a symmetric map which is additive the the trace ä of satisfy the relatio: (x y) C (x, x,,..., x)+c (x, x,,..., x,y) +C (x, x,,..., x,y) ( x x ) ( x )... ( x ) (AA, A,..., A ) ( x ) ( x ) ( x )... ( x ) C r (x,x,...,-r times,y,y,...,r-times) +.+ Now, (A, A,..., A )A C (y, y, y,..., y) ( x ) ( x )... ( x ) x y ad i.e (x y) C (x) C (x, x, x,..., x,y) (A ) (A, A,..., A ) + C (x, x,,..., x,y)+...+ ( x ) ( x ) ( x )... ( x ). C r (x,x,...,-r times,y,y,...,r-times) +.+ C (y), x, y R For symmetric biderivatios satisfies the relatio (x y) (x) (x, y) (y), x,y R For symmetric -derivatios satisfies the relatio (x y)= (x) (x, x, y) (x, y,y) (y), x,y R For symmetric 4-derivatios ä satisfies the relatio: (x y)= (x) 4 (x, x, x,y) 6 (x, x,y,y) 4 (x, y, y,y) (y), x, y R. (AA, A,..., A ) (A, A,..., A )A (A ) (A, A,..., A ). Hece from the above it is clear that the additive map is a symmetric skew reverse derivatio associated with a atiautomorphism. Now let us quote some importat lemmas, which are useful i provig the mai results.. Lemma..6. [] Let be a fixed positive iteger ad R be a! torsio free rig.suppose that y, y, y,..., y R, which satisfy the relatio: ISSN: Page 8

3 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) - Volume 47 Number July 7 y y y... y for =,,,...,.The yi, i. Lemma..7. [4] Let R be a prime rig. Let D: R R be a derivatio ad a R. If a D(x) = or D(x) a =, x R, the we have either a = or D =. Lemma.8. Let R be a prime rig. If a [ x, b ] = or [ x, b ] a =, x R the either a = or b Z(R), the ceter of the rig R. Proof. Replacig x by xy i the relatio a [ x, b ] =, a [ xy, b ] = ax [ y, b ] + a [ x, b ] y = ax [ y, b ] =, x,y R. Thus, a R [ y, b] =, y R. Sice R is prime the either a = or b Z(R).. The mai results: Theorem... Let be ay fixed positive iteger. Let R be a o commutative! torsio free prime rig ad I be ay o zero two sided ideal of R.Suppose that there exist a symmetric skew reverse -derivatio Ä:R R associated with a atiautomorphism á. Let ä deote the trace of Ä such that [ä(x), á(x)] =, for all x I the (x,x,x,..., x ) =, xi R Proof. Our suppositio is that there exist a symmetric skew reverse derivatio Ä: R R associated with a atiautomorphism á such that [ ä(x), á(x) ]=, x I. () Substitutig x by x+ y ( ì ), i (), we have [ ( x + y ), ( x + y)] x, y I [ cr fr ( x, y), ( x ) ( y)]= r [ ( x ) ( y) cr fr ( x, y), ( x ) (y)]= r [ ( x ), ( x )] [ ( x ), (y)] + [ ( y ), ( x )] [ ( y ), ( y )] [ cr fr ( x, y), (x)] r [ cr fr ( x, y), ( y )] r [ ( x ), (y)] [c f ( x,y), ( x )] [c f ( x,y), ( x )] [ c f ( x,y), ( y )] [c f ( x,y), ( x )] [ c f ( x,y), ( y )]... [c f ( x,y), ( y )] [ ( y ), ( x )]. ISSN: -57 Applyig Lemma.6 to equatio (), [ c f ( x,y), ( x )] [ ( x ), (y)]. Now replacig y by yx i equatio (), [ c{ f ( x, x ) y ( x ) f ( x, y)}, ( x )] [ ( x ), (x) ( y )] () c ( x )[ y, ( x )] ( x ) [ c f ( x, y), ( x )] (x)[ ( x), ( y )] ( x ) [ c f ( x, y), ( x )] [ ( x ), ( y )] c ( x )[ y, ( x )] c ( x )[ y, ( x )] Usig!-torsio freeess, ( x)[ y, ( x)], x, y I Now replacig y by yr i equatio (4), ( x)[ yr, ( x)], x, y I r R (4) ( x )[ y, ( x )] r ( x ) y[ r, ( x )] ( x ) y[ r, ( x )], x, y I ad r R (5) Sice R is prime the by usig Lemma.8 to equatio either ( x ), x I \ Z (6) (or) ( x ) Z( R), x I \ Z. Let x I Z, y I ad y Z the y+ x I \ Z From relatio (6), we have, ( y x) cs f s ( y, x ) s s c ( y ) c ( x ) cs f s ( y, x ) s s c ( x ) c s f s ( y, x ) s c f ( y, x ) c f ( y, x ) c f ( y, x )... c f ( y, x ) c ( x ) Agai applyig Lemma.6 to equatio (7), c f ( y, x) f ( y, x) ( y, y, y,..., y, x) c f ( y, x ) f ( y, x) ( y, y, y,..., y, x, x)..., () c f ( y, x) f ( y, x) c ( x ) ( x, x, x,..., x, x ). Page 8 (7)

4 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) - Volume 47 Number July 7 Hece from the above we have, (p, x, x,..., x, x ) x. ( x), x I Z (8) Now for each value of l=,,,..., let us deote (p, x,..., x, x ), p R, xi I Tl ( x) ( x, x, x,..., x, xl, xl, xl,..., x ), where x, xi I, i =l +,l +,..., T ( x ) ( x ), x I Now replacig x by xp, where p R i (5), (9) Let ç ( ç ) be ay positive iteger.the from (9), (p, p, x,..., x, x ), where p, p R, xi I Repeatig the above process we fially obtai, (p, p, p,..., p, p ), pi R Hece the proof of the theorem is completed. Theorem... Let be ay fixed positive iteger. Let R be o commutative! torsio free semiprime rig ad I be ay o zero two sided ideal of R. Suppose that there exist a symmetric skew reverse -derivatio Ä: R R associated with a atiautomorphism á. Let ä deote the trace of Ä such that ä is commutig o I ad [ä(x),á (x)] Z(R), for all x I the [ä(x), á(x)] =, x I. Proof. Our suppositio is that there exist a symmetric skew reverse derivatio Ä: R R associated with a atiautomorphism á such that [ä(x), x] =, for all x I T ( x x ) l T ( x ) T ( x ) cl Tl ( x ) l l ( x ) ( x ) cl Tl ( x ) l l cl Tl ( x ), x, x I l ct ( x ) c T ( x ) ct ( x )... (5) Now applyig Lemma.8 to equatio (5), ad [ä(x), á(x)] Z(R), x I. c T ( x ). () Agai applyig Lemma.8 to equatio (), ct ( x) T( x) ( x, x, x,..., x ) ct ( x) T ( x) ( x, x, x,..., x )..., c T ( x) T ( x) ( x, x, x,..., x, x ) Hece from the above, we have T-(x) =, x I () Agai let ô ( ô -) be ay positive iteger. The from () T ( x x ), x, x I t T ( x ) T ( x ) ct Tt ( x ) t ct ( x ) c T ( x ) ct ( x )... c T ( x ), x I. Agai applyig Lemma () to result () we have, Let ì ( ì ) be ay positive iteger. Substitutig x by x+ y i () we have, [ ( x + y ), ( x + y)] Z, x, y I [ cr fr ( x, y), ( x ) ( y)] Z r [ ( x ), (y)] [ c f ( x,y), ( x )] [c f ( x,y), ( x )] [ c f ( x,y), ( y )]... [c f ( x,y), ( y )] [ ( y ), ( x )] () ( x, x, x,...x, x, x ) [ ( y ), ( y )] [ ( x ), ( x )] Z, x, y I Now commutig (7) with ä(x), [ ( x ), (y)] [ c f ( x,y), ( x )], ( x ) [c f ( x,y), ( x )] [ c f ( x,y), ( y )], ( x )... [c f ( x,y), ( y )] [ ( y ), ( x )], ( x ) [ ( y ), ( y )], ( x ) (8) [ ( x ), ( x )], ( x ), x, y I T ( x), x I () Cotiuig the above process, fially we obtai T(x) =, x I. ( x, x, x,..., x, x ), xi I. (4) Now replacig x by xp, where p R i (4), Applyig Lemma..6 to equatio (8),we have ( xp, x, x,..., x, x ) [c f ( x,y), ( x)] [ ( x ), (y)], ( x ) (p, x, x,..., x, x ) x (p ) ( x, x, x,..., x, x ) ISSN: -57 (7) (9) Now replacig y by x i equatio (9), we get [ c f ( x, x ), ( x )], ( x ) [ ( x ), ( x )], ( x ) Page 84

5 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) - Volume 47 Number July 7 ( x )[ ( x ), ( y )], ( x ) (c ) ( x )[ ( x ), ( x )], ( x ) c[ ( x ), ( x )] y, ( x ) c ( x )[ y, ( x )], ( x ) c ( x )[ f ( x,y), ( x )], ( x ) [ ( x ), ( x )] ( y ), ( x ) ( x )[ ( x ), ( y )], ( x ) [ ( x ), ( x )] ( x ), ( x ) c[ ( x ), ( x )] x, ( x ) c ( x )[ x, ( x )], ( x ) ( c )[ ( x ), ( x )][ ( x ), ( x )] (c ) ( x ) [ ( x ), ( x )], ( x ) [ ( x ), ( x)][ ( x ), ( x )] [ ( x ), ( x )], ( x ) ( x ) c[ ( x ), ( x )][ x, ( x )] c[ ( x ), ( x )], ( x ) x c ( x ) [ x, ( x )], ( x ) c[ ( x ), ( x )][ x, ( x )] ( c )[ ( x ), ( x )] [ ( x ), ( x )] c ( x ) (x),{ x ( x ) ( x ) x} ( c )[ ( x ), ( x )] c ( x ) x,[ (x), ( x )] () ( c )[ ( x ), ( x )], x I Agai commutig equatio (7) with á (x) ad usig Lemma..6, [ c f ( x, y ), ( x )] [ ( x ), ( y )], ( x ) () Now replacig y by yx i the above equatio (), [c f ( x, yx ), ( x )], ( x ) + [ ( x ), ( yx )], ( x ) [c f ( x, x ) y ( x ) f ( x,y), ( x )], ( x ) [ ( x ), ( x ) ( y )], ( x ) c [ ( x ), ( x )], ( x ) y c ( x ) [ y, ( x )], ( x ) c[ ( x ), ( x )] [ y, ( x )] c ( x ) [ f ( x,y), ( x )], ( x ) c[ ( x ), ( x )][ f ( x,y), ( x )] +[ ( x ), ( x )] [ ( y ), ( x )] [ ( x ), ( x )], ( x ) ( y ) ( x ) [ ( x ), ( y )], ( x ) [ ( x ), ( x )] [ ( x ), ( y )] c[ ( x ), ( x )][ y, ( x )] c ( x ) [ y, ( x )], ( x ) +[ ( x ), ( x )] [ ( y ), ( x )]. () Now replacig y by ( x )[ ( x ), ( x )] i relatio (), [ ( x ), ( x )] [ ( x ), ( x )] ( x ), ( x ) c[ ( x ), ( x )] y, ( x ) c ( x )[ y, ( x )], ( x ) ISSN: -57 c ( x ) [ ( x )[ ( x ), ( x )], ( x )], ( x ) [ ( x ), ( x ) ( y )], ( x ) c ( x )[ f ( x,y), ( x )], ( x ) [ ( x ), ( x )] ( y ), ( x ) [c ( x ) y, ( x )], ( x ) [ c ( x ) f ( x,y), ( x )], ( x ) c[ ( x ), ( x )] f ( x,y), ( x ) c[ ( x ), ( x )] ( x )[ ( x ), ( x )], ( x ) c[ ( x ), ( x )] [ y, ( x )] c ( x ) [ ( x ), ( x )], ( x ) c[ ( x ), ( x )] [ ( x ), ( x )] (c )[ ( x ), ( x )], x I () (c )[ ( x ), ( x )] R (c )[ ( x ), ( x )] Sice R is semiprime the, (c ) [ ( x ), ( x )], x I x I. (4) Page 85

6 Iteratioal Joural of Mathematics Treds ad Techology (IJMTT) - Volume 47 Number July 7 Now combiig () ad (4), [ ( x ), ( x )], x I. As the ceter of the semiprime rig cotais o o-zero ilpotet elemets we have [ ä(x), á(x) ] =, for all x I. Hece the proof of the theorem is completed. Theorem... Let be ay fixed positive iteger. Let R be a! torsio free prime rig ad I be ay o zero two sided ideal of R. Suppose that there exist a o zero symmetric skew reverse -derivatio Ä: R R, associated with a atiautomorphism á. Let ä deote the trace of such that ä is commutig o I ad [ä(x), á(x)] Z(R), for all x I the R must be commutative. [9] [] [] [] [] [4] Proof. O cotrary suppose that R is a o commutative prime rig. The from Theorem.., we have [ä(x), á(x)] =, for all x I. Now from Theorem.., we have Ä =, x R, which is a cotradictio. Hece R must be commutative prime rig. [5] [6] [7] Coflict of Iterests: [8] The authors declare that there is o coflict of iterests. [9] [] Refereces: [] [] [] [4] [5] [6] [7] [8] Mohammad Ashraf, O symmetric biderivatios i rigs, Red.Istit.Mat.Uiv, Trieste, (999), 5-6. M. Bresar ad J. Vukma, O some additive mappigs i rigs with ivolutio, Aequatioes. Math. 8 (989), L.O. Chug ad Jiag Luh, Semiprime rigs with ilpotet derivatives, Caad.Math.Bull., 4 (98), Basudeb Dhara ad Faiza Shujat, Symmetric skew -derivatios i prime ad semi prime rigs, Southeast Asia Bull. of Math., -9, (7). Ajda Foser, Prime ad semiprime rigs with symmetric skew derivatios, Aequatioes.Math. 87 (4), 9-. I.N. Herstei.., Jorda derivatios of prime rigs, Proc.Amer.Math.Soc. 8 (957), 4-. Y.S. Jug. ad K.H.Park, O prime ad semiprime rigs with permutig -derivatios, Bull.Korea.Math.Soc., 44 (7), C. Jayasubba Reddy, S.Vasath Kumar ad S.Mallikarjua Rao, ISSN: -57 [] [] Symmetric skew 4- derivatios o prime rigs, Global Jr. of pure ad appl. Math., (6), -8. C. Jayasubba Reddy, V.Vijaya Kumar ad K.Hemavati, Prime rig with symmetric skew -reverse derivatio, IJMCAR, 4 (4), C. Jayasubba Reddy, V.Vijaya Kumar ad S.Mallikarjua Rao, Semiprime rig with Symmetric skew -reverse derivatios, Tras. o Math., (5), -5. C. Jayasubba Reddy, V.Vijaya Kumar ad S.Mallikarjua Rao, Prime rig with symmetric skew 4-reverse derivatio, It. J. of Pure Algebra, 6 (6), C. Jayasubba Reddy, S.Vasath Kumar ad K. Subbarayudu, Symmetric skew 4-reverse derivatios o prime rigs, IOSR Jr. of Math., (6), 6-6. Joseph H Maye, Cetralizig mappigs of prime rigs, Caad.Math.Bull, 7 (98), -6. E.C. Poser, Derivatios i prime rigs, Amer.Math.Soc, 8 (957), 9-. K.H. Park,, O 4-permutig 4-derivatios i prime ad semiprimerigs,. Korea.Soc..Educ.Ser. B.Pure.Appl.Math, 4 (7), K.H. Park, O prime ad semiprime rigs with symmetric derivatios, J.Chugcheog Math.Soc., (9), Faiza Shujat, Abuzaid Asari, Symmetric skew 4-derivatios o prime rigs, J. Math. Compt.Sci., 4 (4), M Samma ad N Alyamai, Derivatios ad Reverse derivatios, It.Math.Forum, (7), J. Vukma, Commutig ad cetralizig mappigs i prime rigs, Proc.Amer.Math.Soc, 9 (99), J. Vukma, Symmetric biderivatios o prime ad semiprime rigs, Aequatioes Math.Soc. 8 (989), J. Vukma, Two results cocerig symmetric biderivatios o prime rigs, Aequatioes. Math, 49 (99), V.K. Yadav. ad R.K. Sharma., Skew -derivatios o prime ad semi prime rigs, A.Uiv.Ferrara, (6), -. Page 86