The purpose of this paper is to prove the following result: Let m 1, n 1 be fixed integers and let R be a (m + n + 2)!-torsion free semiprime ring with the identity element. Suppose there exists an additive mapping T : R R, such that 

T(xm+n+1) = xm T(x) xn 

holds for all x R. In this case T is a centralizer

Irena Kosi-Ulbl

The purpose of this paper is to prove the following result: Let m 1, n 1 be fixed integers and let R be a (m + n + 2)!-torsion free semiprime ring with the identity element. Suppose there exists an additive mapping T : R R, such that 



T(xm+n+1) = xm T(x) xn 



holds for all x R. In this case T is a centralizer

Kosi-Ulbl, Irena

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GLASNIK MATEMATICKIVol. 39(59)(2004), 21 { 26A REMARK ON CENTRALIZERS IN SEMIPRIME RINGSIrena Kosi-UlblUniversity of Maribor, SloveniaAbstract. The purpose of this paper is to prove the following result:Let m  1, n  1 be xed integers and let R be a (m+ n+ 2)!-torsion freesemiprime ring with the identity element. Suppose there exists an additivemapping T : R ! R, such that T (xm+n+1) = xmT (x)xn holds for allx 2 R. In this case T is a centralizer.This research has been motivated by the work of Vukman [7]. Through-out, R will represent an associative ring with center Z(R): A ring R is n-torsion free, where n is an integer, in case nx = 0; x 2 R implies x = 0: Asusual the commutator xy   yx will be denoted by [x; y] : We shall use basiscommutator identities [xy; z] = [x; z]y+ x[y; z] and [x; yz] = [x; y] z+ y [x; z] :Recall that R is prime if aRb = (0) implies a = 0 or b = 0, and is semiprime ifaRa = (0) implies a = 0. An additive mapping D : R! R; where R is an ar-bitrary ring, is called a derivation in caseD(xy) = D(x)y+xD(y) holds for allpairs x; y 2 R and is called a Jordan derivation in caseD(x2) = D(x)x+xD(x)holds for all x 2 R. Obviously, any derivation is a Jordan derivation. Theconverse is in general not true. Herstein [5] proved that any Jordan deriva-tion on a 2-torsion free prime ring is a derivation. A brief proof of Hersteintheorem can be found in [3] : Cusack [4] has extended Herstein theorem to 2-torsion free semiprime rings (see also [1] for an alternative proof). An additivemapping T : R ! R is called a left (right) centralizer in case T (xy) = T (x)y(T (xy) = xT (y)) holds for all x; y 2 R: We follow Zalar [8] and call T a cen-tralizer in case T is both left and right centralizer. In case R has the identityelement T : R ! R is left (right) centralizer i T is of the form T (x) = ax(T (x) = xa) for some xed element a 2 R. An additive mapping T : R! R iscalled a left (right) Jordan centralizer in case T (x2) = T (x)x (T (x2) = xT (x))2000 Mathematics Subject Classication. 16N60, 39B05.Key words and phrases. Prime ring, semiprime ring, left (right) centralizer, left (right)Jordan centralizer, centralizer.2122 I. KOSI-ULBLholds for all x 2 R. Following ideas from [1] Zalar [8] has proved that any left(right) Jordan centralizer on a 2-torsion free semiprime ring is a left (right)centralizer. Recently Vukman [6] has proved that in case T : R ! R is anadditive mapping, where R is a 2-torsion free semiprime ring, which satisesthe identity 2T (x2) = T (x)x+xT (x) for all x 2 R, then T is a centralizer. In[7] Vukman set the following conjecture:Let R be a semiprime ring with suitable torsion restrictions. Supposethere exists an additive mapping T : R ! R, such that T (xm+n+1) =xmT (x)xn holds for all x 2 R, where m  1, n  1 are some integers.In this case T is a centralizer.The result below gives an armative answer to the question above in caseR has the identity element.Theorem 1. Let m  1, n  1 be xed integers and let R be a(m+ n+ 2)!-torsion free semiprime ring with the identity element. Sup-pose there exists an additive mapping T : R ! R, such that T (xm+n+1) =xmT (x)xn holds for all x 2 R. In this case T is a centralizer.Let us see the background of the conjecture and the theorem above. Anadditive mapping D : R ! R, where R is an arbitrary ring, is called Jordantriple derivation, in case D(xyx) = D(x)yx + xD(y)x + xyD(x) holds forall pairs x; y 2 R. One can easily prove that any Jordan derivation on anarbitrary ring is a Jordan triple derivation (see [3]). Bresar [2] has proved thatany Jordan triple derivation on a 2-torsion free semiprime ring is a derivation.This result inspired Vukman [7] to prove the following result: Let T : R! Rbe an additive mapping, where R is a 2-torsion free semiprime ring. Supposethat(1) T (xyx) = xT (y)xholds for all x 2 R: In this case T is a centralizer. For y = x the identity (1)reduces to(2) T (x3) = xT (x)x; x 2 R:The question arises whether the identity (2) on a 2-torsion free semiprimering implies that T is a centralizer. Vukman [7] proved that the answer tothis question is armative in case R has the identity element. The identity(2) leads to the conjecture above.Proof of Theorem 1. We have the relation(3) T (xm+n+1) = xmT (x)xn; x 2 R:A REMARK ON CENTRALIZERS IN SEMIPRIME RINGS 23Replacing in the above relation x + e for x; where e denotes the identityelement, we obtainm+n+1Xi=0m+ n+ 1iT xm+n+1 i== mXi=0mixm i!(T (x) + a) nXi=0nixn i!; x 2 R;(4)where a stands for T (e) : Using (3) and rearranging the equation (4) in senseof collecting together terms involving equal number of factors of e we obtain:m+ n+ 11T xm+ne m0n1xmT (x)xn 1e m0n0xmaxn m1n0xm 1T (x) xne+m+ n+ 12T xm+n 1e2 m0n2xmT (x)xn 2e2 m0n1xmaxn 1e m1n1xm 1T (x)xn 1e2 m1n0xm 1axne m2n0xm 2T (x)xne2+   ++m+ n+ 1m+ nT xem+n  mm  1nnxaem+n 1 mmnnT (x) em+n  mmnn  1axem+n 1 = 0; x 2 R;or shortly(5)m+nXi=1fi (x; e) = 0; x 2 R;where fi (x; e) stands for the expression of terms involving i factors of e:Replacing e by 2e; 3e; : : : ; (m+ n) e in turn in the equation (5), and ex-pressing the resulting system of m + n homogeneous equations, we see thatthe coecient matrix of the system is a van der Monde matrix266641 1    12 22    2m+n............m+ n (m+ n)2    (m+ n)m+n37775 :24 I. KOSI-ULBLSince the determinant of the matrix is dierent from zero, it follows that thesystem has only a trivial solution.In particular,fm+n 1 (x; e) =m+ n+ 1m+ n  1T x2  mm  2x2a nn  2ax2 mm  1xT (x) nn  1T (x) x mm  1nn  1xax= 0; x 2 Randfm+n (x; e) =m+ n+ 1m+ nT (x) mm  1xa nn  1ax mmnnT (x)= 0; x 2 R:Since R is a (m+ n+ 2)!-torsion free ring, the above equations reduce to(m+ n+ 1) (m+ n)T x2= m (m  1)x2a+ n (n  1) ax2 + 2mnxax+2mxT (x) + 2nT (x) x; x 2 R(6)and(7) (m+ n)T (x) = mxa+ nax; x 2 R;respectively.According to (7) one obtains the relation(m+ n)T x2= mx2a+ nax2; x 2 R:Using the above connection one can replace the expression (m+ n)T x2with mx2a+ nax2 in the relation (6). Thus we have(m+ n+ 1) mx2a+ nax2= m (m  1)x2a+ n (n  1) ax2+2mnxax+ 2mxT (x) + 2nT (x) xFrom the above relation we obtain(2m+mn)x2a+ (2n+mn) ax2   2mnxax 2mxT (x)  2nT (x) x = 0; x 2 R:(8)Rearranging the above relation gives2m x2a  xT (x)+ 2n  ax2   T (x) x+mn x2a+ ax2   2xax = 0; x 2 R:(9)A REMARK ON CENTRALIZERS IN SEMIPRIME RINGS 25Left and right multiplication of the relation (7) by x gives(10) (m+ n)xT (x) = mx2a+ nxax; x 2 Rand(11) (m+ n)T (x) x = mxax+ nax2; x 2 R;respectively.Multiplication of the relation (9) by (m+ n) e gives2m (m+ n)x2a  (m+ n)xT (x)+ 2n  (m+ n) ax2   (m+ n)T (x) x+mn (m+ n) x2a+ ax2   2xax = 0; x 2 R:(12)Using (10) and (11) in the relation (12) one obtains0 = 2m (m+ n)x2a mx2a  nxax+ 2n  (m+ n) ax2  mxax  nax2+mn (m+ n) x2a+ ax2   2xax= 2m (m+ n)  2m2 +mn (m+ n)x2a+ 2n (m+ n)  2n2 +mn (m+ n) ax2   (4mn+ 2mn (m+ n)) xax= mn (m+ n+ 2)x2a+mn (m+ n+ 2) ax2 2mn (m+ n+ 2)xax; x 2 R:Since R is a (m+ n+ 2)!-torsion free ring we obtain(13) x2a+ ax2   2xax = 0; x 2 R:The above relation can be written in the form(14) [[a; x] ; x] = 0; x 2 R:The rest of the proof goes through in the same way as in the end of the proofof Theorem 2 in [7], but we proceed for the sake of completeness. Puttingx+ y for x in the above relation we obtain(15) [[a; x] ; y] + [[a; y] ; x] = 0; x; y 2 R:Putting xy for y in relation (15) we obtain because of (14) and (15):0 = [[a; x] ; xy] + [[a; xy] ; x]= [[a; x] ; x] y + x [[a; x] ; y] + [[a; x] y + x [a; y] ; x]= x [[a; x] ; y] + [[a; x] ; x] y + [a; x] [y; x] + x [[a; y] ; x]= [a; x] [y; x] ; x; y 2 R:Thus we have[a; x] [y; x] = 0; x; y 2 R:The substitution ya for y in the above relation gives(16) [a; x] y [a; x] = 0; x; y 2 R:26 I. KOSI-ULBLLet us point out that so far we have not used the assumption that Ris semiprime. Since R is semiprime, it follows from the relation (16) that[a; x] = 0; x 2 R: In other words, a 2 Z (R) ; which reduces the relation (7)to T (x) = ax; x 2 R: The proof of the theorem is complete.Acknowledgements.I would like to thank to Professor Joso Vukman for helpful suggestions.References[1] M. Bresar, Jordan derivations on semiprime rings, Proc. Amer., Math. Soc. 104 (1988),1003-1006.[2] M. Bresar, Jordan Mappings of semiprime rings, J. Algebra 127 (1989), 218-228.[3] M. Bresar and J. Vukman, Jordan derivations on prime rings, Bull. Austral. Math.Soc. 3 (1988), 321-322.[4] J. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), 1104-1110.[5] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957),1104-1110.[6] J. Vukman, An identity related to centralizers in semiprime rings, Comment. Math.Univ. Carolinae 40 (1999), 447-456.[7] J. Vukman, Centralizers of semiprime rings, Comment. Math. Univ. Carolinae, 42(2001), 237-245.[8] B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carolinae 32(1991), 609-614.Department of MathematicsUniversity of MariborPEF, Koroska c. 160, 2000 MariborSloveniaE-mail : irena.kosi@uni-mb.siReceived : 03.02.2003Revised : 19.03.2003

A remark on centralizers in semiprime rings

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