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An Identity Related to Centralizers in Semiprime Rings .. 449 \ Hspacetheorem∗{\ Thatf I L D L } =An 0 Comma Identity

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An Identity Related to Centralizers in Semiprime Rings .. 449 \ Hspacetheorem∗{\ Thatf I L D L } =An 0 Comma Identity An identity related to centralizers in semiprime rings .. 449 n hspacetheorem∗fn thatf i l D l g =An 0 comma identity .. since related we have to assumed centralizers that R is noncommutative in semiprime period rings .. Innquad 449 other words in this case we have a in Z open parenthesis R closing parenthesis period .. But we have the weaker assumption nnoindentthat R is atheoremthat semiprime ring and $D in semiprime = 0 rings ,$ Posnernquad quoterightsince s second we have theorem assumed in that $ R $ is noncommutative . nquad In othergeneral words does not in hold this as the case following we have simple $ example a n in showsZ period ( .. Take R R ) sub 1 . to $ ben aquad But we have the weaker assumption thatnoncommutative $ R $ is prime a semiprime ring and R sub ring 2 to and be a in commutative semiprime prime rings ring that Posner admits ' s second theorem in generala nonzero does derivation not d hold : R sub as 2 the right following arrow R sub simple 2 period Thenexample R sub shows 1 times . Rn subquad 2 isTake a noncommutative $ R f 1 semiprimeg$ to be a noncommutativering and the mapping prime D : R ring sub 1 and times $ R R subf 2 right2 g$ arrow to R be sub a 1 commutative times R sub 2 comma prime D ring open parenthesis that admits r sub 1 comma a nonzero derivation $ d : R f 2 g nrightarrow R f 2 g . $ Then $ R f 1 g ntimes r sub 2 closing parenthesis = open parenthesis 0 comma dAn open identity parenthesis related to rcentralizers sub 2 closing in semiprime parenthesis rings closing449 parenthesis is a Rnonzerof 2 g$theorem is a that noncommutativeD = 0; since semiprime we have assumed that R is noncommutative . In other words in ringderivation andthis thewhich case mapping mapswe have R suba 2 $D 1Z( timesR): : RBut sub R we 2f intohave1 itsg the center nweakertimes period assumptionR In thef 2 sequelthatgR nweisrightarrow a shall semiprime prove that ringR andf in1 g ntimes R f 2 g ,in D case we (semiprime have r anf rings inner1 g Posner derivation,' r s second Df comma2 g theorem) which = in is general commuting ( does 0 on not , a hold noncommutative d as ( the following r f 2 simpleg ) example )$ is anonzero derivation which maps $ R f 1 g ntimes R f 2 g$ into its center . In the sequel we shall prove that semiprimeshows ring . comma Take thenR1 to D be = 0 a period noncommutative For this purpose prime we ring put and x plusR2 to y forbe a x commutative in open parenthesis prime 3ring closing that parenthesis period Wein have caseadmits we have a nonzero an inner derivation derivationd : R ! R : $Then DR ,× $R whichis a noncommutative is commuting semiprime on a ring noncommutative and the semiprime ring , then $D =2 02 . $ For1 this2 purposeweput $x + y$ for $x$ in ( 3 ) .Wehave brackleftmapping D open parenthesisD : R1 × R2 x! closingR1 × R parenthesis2;D(r1; r2) comma = (0; d( yr2 brackright)) is a nonzero plusbrackleft derivation D which open mapsparenthesisR1 × R y2 closing parenthesis comma x brackrightinto its center = 0 comma . In the x sequelcomma we y in shall R comma prove that and in in case particular we have for an y = inner a comma derivation sinceD; Dwhich open parenthesis is a closing parenthesisnnoindentcommuting = 0$[D(x),y]+[D(y),x]=0,x comma on a noncommutative semiprime ring , then D = 0: For this purpose we put x + y for x ,we y obtainninin brackleft ( 3 )R . We D have,$ open parenthesisandinparticularfor x closing parenthesis $y comma = a brackright a ,$ = 0 since comma x $D in R period ( In a other ) words = we 0 have, $ [D(x); y] + [D(y); x] = 0; x; y 2 R; and in particular for y = a; since D(a) = 0; Equation:we open obtain parenthesis [D(x); a] = 4 closing 0; x 2 R: parenthesisIn other words .. D to we the have power of 2 open parenthesis x closing parenthesis = 0 comma x in Rnnoindent period weobtain$[ D ( x ) , a ] = 0 , x n in R . $ In other words we have Posner quoteright s first theorem brackleft 6 brackright .. asserts that if R is a prime ring of characteristic n begindifferentf a l ifrom g n ∗g two and D comma G are nonzeroD derivations2(x) = 0; on x R2 commaR: then DG cannot be (4) n taga derivation∗f$ ( period 4 ) .. $ Letg D us ^ pointf 2 outg that( Posner x ) quoteright = 0 s first , theorem x n doesin notR. hold for nendsemiprimef a l i gPosner n ∗g rings ' period s first .. theorem However [ 6 comma ] asserts it is wellthat known if R is that a prime if D ring and ofDto characteristic the power of different 2 are derivations from two in 2 hyphenand torsionD; G freeare semiprime nonzero derivations ring comma on thenR; then D = 0DG opencannot parenthesis be a derivation see the proof . of Let Lemma us point 1 period out that 1 period 9 in brackleft 5nnoindent brackrightPosner closingPosner ' parenthesiss first ' s theorem first period does theorem not hold [ for 6 ] nquad asserts that if $ R $ is a prime ring of characteristic 2 differentHence wesemiprime have from D = two rings 0 period and . However .. $D In other , it , words is well G$ a knownin Z are open that nonzero parenthesis if D and derivationsD R closingare derivations parenthesis on in $R period , .. $ Now then it follows $DG$ from open cannot be parenthesisa derivation2 2 - closing torsion . parenthesisn freequad semiprimeLet that us ring point , then outD = that 0( see Posner the proof ' of s Lemma first 1 . theorem 1 . 9 in [ 5 does ] ) . not hold for T open parenthesisHence we have x closingD = 0 parenthesis: In other = words ax commaa 2 Z( andR): T openNow itparenthesis follows from x closing ( 2 ) that parenthesis = xa for all x in R comma whichnnoindent completesT (semiprimex) =theax; proofand of ringsT ( thex) = theoremxa . nforquad all xHowever2 R; which , completes it is well the proof known of the that theorem if in $D$ case R has and an $D^f 2 g$ are derivations in in case Ridentity has an element identity . element period nnoindentIn the sequelIn2 the− we sequeltorsion present we present the free proof the semiprime of proof Theorem of Theorem 1 ring without 1, without then assuming assuming $D that R that = hasR 0has an ( identity $ see element the proof . The of Lemma1 . 1 . 9 in [ 5 ] ) . an identityproof element is , as we period shall The see , proof rather is long comma but itas is we elementary shall see commaP parenleft rather− longifive but−nt it−oo is− elementaryhe sense h−te− e e−brackleftT −ccparenleft−ia−fi−x nnoindentP parenleft-ihatHencewehave−r five-nit −− t-orequires o-h subno e$Dsp sense h-t = e-h 0 sub a .$ t-r itn toquad the powerInparenright other of e− endashc k− wordsperiod requires− $xbrackright a no spn toin−noequal the powerZ−w ( of e-brackleft R ) T-c . $ cnquad parenleft-iNowl a-fi-xe it− zerocomma follows sub parenright-c− fromdge concerning k-period-x( 2 ) that semiprime sub brackright-n rings in o ord equal-wW − le e-zeror t n − comma-do − it − f sube−on ge− concerningl l o − dw − semiprime rings in ord W-ett r− t n-o-ioh − t-fpr− subeo − e-op − n-lv lr o-do − w-teo − t-oth h-p− f: sub r-e o-p-v r o-e o-t h-f sub period dertoa nnoindentt sub olrtolr o o$To r o aa r aa sub (x x in2 xR: R periodI )n o =I n ochie to ax the power ,$and$T of derto a chie ( x ) = xa$ forall $x n in Ru .. , t $ .. a which .. T s .. completes h r l t the proof ofu the t a theoremT s h r l t inopen case parenthesis $ R $ 6 closing has an parenthesis identity element . Sin .. r to the power of i-t ten n .. th e sub f (6) nnoindenti P o sub uIn tin the gin t sequel e el .. io we n open present parenthesis the 1 proof closing of parenthesis Theorem x 1 r sub without plus y f assuming o sub or to that the power $ R of $n x nhas i to the an identity element . The proof is , as we shall see , rather long but it is elementary power of n sub e btaii−t s .. o mm u .. n n n $Pf p a rSin e n l er f t −iteng nf i th v ee−fni P o tu−tino gin o− th e elf ioe ng$ (1)xr sense+y f oorx $n h−ietbtai e s−h o mmf a u tn−r g i t ^f e g −−f r e q u i r e s g$ t .. h .. r .. t t h r t noEquation: $ sp ^f opene−b parenthesis r a c k l e f t to the T−c power cf ofp a7 r closing e n l e f t parenthesis−i g a−f .. i − xx tof theparenright power of closing−c gg parenthesisk−period y plus−x Tf subbrackright open −n parenthesiso equal plus−w g closing$ l parenthesis $ e−zero plus comma−d f ge g$ concerning semiprime rings in ord $ W−e $ r t $ n−o−i t−f f e−o g n−l $ l $ o−d w−t t−o h−p f r−e g o−p−v $ x)y + T + )+ (7) r $ o−e o−t h−f f .
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