240 .. J period Vukman nnoindentLine 1 C-F-w-L240 n o-h-o-e-rquad J t-m-i . Vukman sub c-m y-h-b h-g-n-b to the power of e-i-i sub n-v-e y-e-g-a s-brackleft b-o-one-parenleft sub three-x-v-e c-e-parenright-parenright comma-a r-w-u-x-e l-i-brackright-s t-e-a i-o-t-n o-parenleft-f-parenleft parenleft-one-n to the powern [ n begin of four-one-sixf a l i g n e d parenright-one-ag C−F−w−L on-period-parenright-w−h−o−e−r t−m−i Linef 2c open−m g parenthesisy−h−b 1 7 h closing−g−n−b parenthesis ^f e−i− bracklefti f n− Tv− parenleft-xe y−e−g−a closings−b r a parenthesisc k l e f t gg commab−o− xone brackright−p a r e n x l e y f brackleftt f three parenleft-T−x−v−e x closing c−e− parenthesisparenright comma−parenright x brackright x comma = 0 comma−a x r− commaw−u−x−e yl− subi−brackright in R sub period−s t−e−a g i−o−t−n o−p a r e n l e f t −f−parenleft parenleft −one−n ^f four−one−s i x parenrightRi g-h t m− utone p− la c-ig a t-in− onperiod .. f open−parenright parenthesis− 1w closingnn parenthesis b y brackleft T open parenthesis x comma-parenright x (open 1 parenthesis 7 ) 1 8 closing [ T parenthesis parenleft brackleft-brackleft−x ) , T open x parenthesis ] x x comma y [ x comma parenleft x x brackleft−T T x open ) parenthesis , x x] closing x parenthesis = 0 comma , x sub ,x brackright y f n Rowin g 1 commaR f Row. gn 2end 0 .f ina lR i g period n e d g n ] 240 J . Vukman Sub t-r ac i-t ng open parenthesis 18 closing parenthesis fro m open parenthesis 7 closing parenthesis o neo b-t a n s brackleft brackleft T open parenthesis x comma-parenright x comma x brackright y n centerline fRi $ g−h $ t m ut p l $ c−i $ a $ t−i $ on nquad f(1)by$[ T ( x e−i−in−v−ey−e−g−as−brackleft four−one−sixparenright−one−a a to theC power− F of− w n ..− dLo ..− rh i− too the− e power− rt − ofm m− eic ..−m ny ..− Rh ..− hbh ..− pg − n − b b − o − one − parenleftthree−x−v−ec−e−parenright−parenrightcomma−ar−w−u−x−el−i−brackright−st−e−ai − o − t − no − parenleft − f − parenleftparenleft − one − n n − period − parenright − w commaopen−parenright parenthesis x open x parenthesis $ g (17) [T parenleft − x); x]xy[parenleft − T x); x]x = 0; x; y R T .. h e .. open parenthesis 9 to the power of closing parenthesis g .. v 2 : n [( 1 8 ) brackleft −brackleft T ( x , x , x x [ T ( x ) , f x g Line 1 brackleft to theRi g power− h t of m brackleft ut p l c − subi a Tt − openi on parenthesis f ( 1 ) b ysub [T ( closingxcomma parenthesis− parenrightx to the power of x comma brackright y] brackright =nbegin plusf array brackleftgf cg brackleft, nn 0 subnend T xf toarray theg powery n in of closingR. parenthesisn ] y brackright comma x plus brackleft brackleft sub T open parenthesis Line 2 open parenthesis 2 0 closing parenthesis comma y; brackright plus brackleft sub brackleft T open (18) brackleft − brackleftT (x; x; xx[T (x); ] = y 2 R: parenthesis sub x closing parenthesis comma y to the power of brackright commax x0 brackright plus sub brackleft brackleft T open parenthesisnnoindent toSub the power $ t− ofr y $ comma ac sub $ i x− brackrightt$ ng(18) = 0 comma from(7)oneo $b−t $ a n s $ [ [ T Sub t − r ac i − t ng ( 18 ) fro m ( 7 ) o neo b − t a n s [[T (xcomma − parenrightx; x]y an d r (P-w x sub comma i h-t-t− tparenright sub ngh e sub ab x o to the , power x of ] x v-f y$ to the power of e-o r r l sub itn 0 to the power of w-parenright e a riv im e n R h p i a-n$ sub a ^ g-tf n openg$ parenthesisnquad d 1n closingquad parenthesisr $ i ^f commam g$ open e parenthesisnquad n n toquad the power$ R of $ parenright-ninenquad h nquad a n d openp parenthesis 2 0 closing parenthesis w eo .. b sub t a i sub n ( x ( n [Line ( 1 x plus brackleft ( n ] T open parenthesis x closing parenthesis comma x y x plus open parenthesis closing parenthesis y brackright Line 2 plus brackleft brackleft xT open parenthesisT y closing h e parenthesis (9) g v comma sub x brackright x-comma x brackright = to the power of x brackleft sub brackleft n centerline fT nquad h e nquad $ ( 9 ^f ) g$ g nquad) v g y] + [[T x y]; x + [[T ( ] y n [ n begin f a l i g n e d g [ ^f [ g f (20)T g ; y(] + ^f [x[T (gx)f; y ); x]g +,[ [T ( ];x ] y = 0; ] + [ [ f T g x ^f ) g y ] , x + [ f [ g f T(x gnne−or w−parenrighte parenright−nine P − wih − t − ttngheabov − f r litn0 a riv i a − ng−t(1); ( a n d ( 2( 2 0 0 ) w ) eo ,b a yi ] + [ f [ g T( f x g ) , y ^f ] g , x ] + f [ g [ T ( ^f y g , t f nx g ] = 0 , nendf a l i g n e d g n ] +[T (x); xyx + ()y] x[[ n hspace ∗fn f i l l g $ P−w f i g h−t−+[[txT ( ty);xf]xngh− commaxg e] ^f =x g f ab o g v−f ^f e−o r g$ r $ l f i t n g 0 ^f w−parenright e g$ a r i v i $ a−n f g−t g (f 1 g ) , ( ^f parenright −nine g$ and(20)weo nquad $ b f t g$ a $ i f n g$ n [ n begin f a l i g n e d g +[T(x),xyx+()y] nn + [ [ f xT g ( y )f , g f x g ] x−comma x ] = ^fxf [ g f [ ggnendf a l i g n e d g n ] Centralizers on semiprime rings .. 241 CentralizersWe have therefore on semiprimebrackleft T open rings parenthesisnquad x241 closing parenthesis comma x brackright yx to the power of 2 minus x to the powerWehavetherefore of 2 y brackleft T open $[ parenthesis T ( x closing x parenthesis ) , commax ] x brackright yx^f 2 plusg xyx − brackleftx ^f 2 Tg openy parenthesis [ T x closing ( x parenthesis) ,x]+xyx[T(x) comma x brackright minus brackleft T open parenthesis ,x] x closing parenthesis− [T(x) comma x brackright xyx ,x = 0 comma ] xyx =x 0comma , y$ in R comma which reduces because of open parenthesis 9 closing parenthesis and open parenthesis 1 5 closing parenthesis to nnoindentbrackleft T open$ x parenthesis , y x closingn in parenthesisR , $ comma which x brackright reduces yx because to the power of ( of 9 2 minus ) and x to ( the 1 5 power ) to of 2 y brackleft T open parenthesis x closing parenthesis comma x brackright = 0 comma x comma y in R period n [[T(x)Left multiplication of the above ,x relation ] by x yx^ givesf 2 g − x ^f 2 g y [ T ( x ) , x ] Centralizers on semiprime rings 241 We have therefore [T (x); x]yx2 − x2y[T (x); x] + xyx[T (x); x] − =x brackleft0 , T x open , parenthesis y n in x closingR. parenthesisn ] comma x brackright yx to the power of 2 minus x to the power of 3 y [T (x); x]xyx = 0; brackleft T open parenthesis x closing parenthesis comma x brackright = 0 comma x comma y in R period x; y 2 R; which reduces because of ( 9 ) and ( 1 5 ) to One .. can .. replace .. in .. the .. above .. relation comma .. according .. to .. open parenthesis 1 5 closing parenthesis comma xnnoindent brackleft T openLeft parenthesis multiplication x closing parenthesis of the above comma relation x brackright by yx .. $ by x $ gives [T (x); x]yx2 − x2y[T (x); x] = 0; x; y 2 R: xy brackleft T open parenthesis x closing parenthesis comma x brackright x comma which gives Equation: open parenthesis 2 1n [x closing parenthesis [T(x)Left multiplication .. xy brackleft of the T ,xabove open relation parenthesis ] by yx^x xgives closingf 2 g parenthesis − x ^f comma3 g xy brackright [ T x to the ( power x of ) 2 minus , x to the] power = of 0 3 y ,brackleft x T , open y parenthesisn in xR. closingn parenthesis] comma x brackright = 0 comma x comma y in R period Left multiplication of the above relationx[T (x by); xT]yx open2 − parenthesisx3y[T (x); x] x = closing 0; x; parenthesis y 2 R: gives Equation: open parenthesis 22 closing parenthesis .. T open parenthesis x closing parenthesis xy brackleft T open parenthesis One can replace in the above relation , according to (15); x[T (x); x]yx by xnnoindent closing parenthesisOne nquad commacan x brackrightnquad r x e top l athe c e powernquad of 2in minusnquad T openthe parenthesisnquad above x closingnquad parenthesisr e l a tx i o to n the , powernquad ofaccording 3 y nquad to nquad brackleft$(15) T open parenthesis ,x[T(x) x closing parenthesis comma x brackright ,x]yx$ = 0 comma x commanquad y inby R period The substitution T open parenthesis x closing parenthesisxy[ yT ( forx); y x] inx; open whichgives parenthesis 2 1 closing parenthesis leads to n beginEquation:f a l i g open n ∗g parenthesis 23 closing parenthesis .