IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation58218110.1155/2009/582181582181Research ArticleVanishing Power Values of Commutators with Derivations on Prime RingsDharaBasudebBellHoward1Department of MathematicsBelda CollegeBeldaPaschim Medinipur 721424 Indiabeldacollege.org200904012010200930082009141220092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let R be a prime ring of char R2, d a nonzero derivation of R and ρ a nonzero right ideal of R such that [[d(x),x]n,[y,d(y)]m]t=0 for all x,yρ, where n0, m0, t1 are fixed integers. If [ρ,ρ]ρ0, then d(ρ)ρ=0.

1. Introduction

Throughout this paper, unless specifically stated, R always denotes a prime ring with center Z(R) and extended centroid C, Q the Martindale quotients ring. Let n be a positive integer. For given a,bR, let [a,b]0=a and let [a,b]1 be the usual commutator ab-ba, and inductively for n>1, [a,b]n=[[a,b]n-1,b]. By d we mean a nonzero derivation in R.

A well-known result proven by Posner  states that if [[d(x),x],y]=0 for all x,yR, then R is commutative. In , Lanski generalized this result of Posner to the Lie ideal. Lanski proved that if U is a noncommutative Lie ideal of R such that [[d(x),x],y]=0 for all xU,yR, then either R is commutative or char R=2 and R satisfies S4, the standard identity in four variables. Bell and Martindale III  studied this identity for a semiprime ring R. They proved that if R is a semiprime ring and [[d(x),x],y]=0 for all x in a non-zero left ideal of R and yR, then R contains a non-zero central ideal. Clearly, this result says that if R is a prime ring, then R must be commutative.

Several authors have studied this kind of Engel type identities with derivation in different ways. In , Herstein proved that if char R2 and [d(x),d(y)]=0 for all x,yR, then R is commutative. In , Filippis showed that if R is of characteristic different from 2 and ρ a non-zero right ideal of R such that [ρ,ρ]ρ0 and [[d(x),x],[d(y),y]]=0 for all x,yρ, then d(ρ)ρ=0.

In continuation of these previous results, it is natural to consider the situation when [[d(x),x]n,[y,d(y)]m]t=0 for all x,yρ, n,m0,t1 are fixed integers. We have studied this identity in the present paper.

It is well known that any derivation of a prime ring R can be uniquely extended to a derivation of Q, and so any derivation of R can be defined on the whole of Q. Moreover Q is a prime ring as well as R and the extended centroid C of R coincides with the center of Q. We refer to [6, 7] for more details.

Denote by Q*CC{X,Y} the free product of the C-algebra Q and C{X,Y}, the free C-algebra in noncommuting indeterminates X,Y.

2. The Case: <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M77"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math></inline-formula> Prime Ring

We need the following lemma.

Lemma 2.1.

Let ρ be a non-zero right ideal of R and d a derivation of R. Then the following conditions are equivalent: (i) d is an inner derivation induced by some bQ such that bρ=0; (ii) d(ρ)ρ=0 (for its proof refer to [8, Lemma]).

We mention an important result which will be used quite frequently as follows.

Theorem 2.2 (see Kharchenko [<xref ref-type="bibr" rid="B11">9</xref>]).

Let R be a prime ring, d a derivation on R and I a non-zero ideal of R. If I satisfies the differential identity f(r1,r2,,rn,d(r1),d(r2),,d(rn))=0for  any  r1,r2,,rnI, then either (i) I satisfies the generalized polynomial identity f(r1,r2,,rn,x1,x2,,xn)=0, or (ii) d is Q-inner, that is, for some qQ,d(x)=[q,x] and I satisfies the generalized polynomial identity f(r1,r2,,rn,[q,r1],[q,r2],,[q,rn])=0.

Theorem 2.3.

Let R be a prime ring of char R2 and d a derivation of R such that [[d(x),x]n,[[y,d(y)]m]t=0 for all x,yR, where n0,m0,t1 are fixed integers. Then R is commutative or d=0.

Proof.

Let R be noncommutative. If d is not Q-inner, then by Kharchenko's Theorem  g(x,y,u,v)  =[[u,x]n,[y,v]m]t=0, for all x,y,u,vR. This is a polynomial identity and hence there exists a field F such that RMk(F) with k>1, and R and Mk(F) satisfy the same polynomial identity [10,Lemma 1]. But by choosing u=e12,  x=e11,  v=e11 and y=e21, we get 0=[[u,x]n,[y,v]m]t  =(-1)tn(e11+(-)te22), which is a contradiction.

Now, let d be Q-inner derivation, say d=ad(a) for some aQ, that is, d(x)=[a,x] for all xR, then we have [[a,x]n+1,[y,[a,y]]m]t=0, for all x,yR. Since d0, aC and hence R satisfies a nontrivial generalized polynomial identity (GPI). By , it follows that RC is a primitive ring with H=Soc(RC)0, and eHe is finite dimensional over C for any minimal idempotent eRC. Moreover we may assume that H is noncommutative; otherwise, R must be commutative which is a contradiction.

Notice that H satisfies [[a,x]n+1,[y,[a,y]]m]t=0 (see [10, Proof of Theorem 1]). For any idempotent eH and xH, we have 0=[[a,e]n+1,[ex(1-e),[a,ex(1-e)]]m]t. Right multiplying by e, we get 0=[[a,e]n+1,[ex(1-e),[a,ex(1-e)]]m]te=[[a,e]n+1,[ex(1-e),[a,ex(1-e)]]m]t-1·{[a,e]n+1([ex(1-e),[a,ex(1-e)]]m)e-([ex(1-e),[a,ex(1-e)]]m)[a,e]n+1e}    =[[a,e]n+1,[ex(1-e),[a,ex(1-e)]]m]t-1  ·{[a,e]n+1(j=0m(-1)j(mj)[a,ex(1-e)]jex(1-e)[a,ex(1-e)]m-j)e-(j=0m(-1)j(mj)[a,ex(1-e)]jex(1-e)[a,ex(1-e)]m-j)[a,e]n+1e}=[[a,e]n+1,[ex(1-e),[a,ex(1-e)]]m]t-1·{0-(j=0m(-1)j(mj)(-ex(1-e)  a)jex(1-e)(aex(1-e))m-j)ae}=-[[a,e]n+1,[ex(1-e),[a,ex(1-e)]]m]t-1(j=0m(mj)(ex(1-e)a)m+1)e=-2m[[a,e]n+1,[ex(1-e),[a,ex(1-e)]]m]t-1(ex(1-e)a)m+1e=(-)t2mt(ex(1-e)a)(m+1)te.

This implies that 0=(-)t2mt((1-e)aex)(m+1)t+1. Since char R2, ((1-e)aex)(m+1)t+1=0. By Levitzki's lemma [12, Lemma 1.1], (1-e)aex=0 for all xH. Since H is prime ring, (1-e)ae=0, that is, eae=ae for any idempotent eH. Now replacing e with 1-e, we get that ea(1-e)=0, that is, eae=ea. Therefore for any idempotent eH, we have [a,e]=0. So a commutes with all idempotents in H. Since H is a simple ring, either H is generated by its idempotents or H does not contain any nontrivial idempotents. The first case gives aC contradicting d0. In the last case, H is a finite dimensional division algebra over C. This implies that H=RC=Q and aH. By [10,Lemma 2], there exists a field F such that HMk(F) and Mk(F) satisfies [[a,x]n+1,[y,[a,y]]m]t. Then by the same argument as earlier, a commutes with all idempotents in Mk(F), again giving the contradiction aC, that is, d=0. This completes the proof of the theorem.

Theorem 2.4.

Let R be a prime ring of char R2, d a non-zero derivation of R and ρ a non-zero right ideal of R such that [[d(x),x]n,[y,d(y)]m]t=0 for all x,yρ, where n0,m0,t1 are fixed integers. If [ρ,ρ]ρ0, then d(ρ)ρ=0.

We begin the proof by proving the following lemma.

Lemma 2.5.

If d(ρ)ρ0 and [[d(x),x]n,[y,d(y)]m]t=0 for all x,yρ,m,n0,t1 are fixed integers, then R satisfies nontrivial generalized polynomial identity (GPI).

Proof.

Suppose on the contrary that R does not satisfy any nontrivial GPI. We may assume that R is noncommutative; otherwise, R satisfies trivially a nontrivial GPI. We consider two cases.Case 1.

Suppose that d is Q-inner derivation induced by an element aQ. Then for any xρ,[[a,xX]n+1,[xY,[a,xY]]m]t is a GPI for R, so it is the zero element in Q*CC{X,Y}. Expanding this, we get ([a,xX]n+1j=0m(-1)j(mj)[a,xY]jxY[a,xY]m-j-j=0m(-1)j(mj)[a,xY]jxY[a,xY]m-j[a,xX]n+1)A(X,Y)=0, where A(X,Y)=[[a,xX]n+1,[xY,[a,xY]]m]t-1. If ax and x are linearly C-independent for some xρ, then ((axX)n+1j=0m(-1)j(mj)[a,xY]jxY[a,xY]m-j-j=0m(-1)j(mj)(axY)jxY[a,xY]m-j[a,xX]n+1)A(X,Y)=0. Again, since ax and x are linearly C-independent, above relation implies that (-xY[a,xY]m[a,xX]n+1)A(X,Y)=0, and so (-xY(axY)m(axX)n+1)A(X,Y)=0. Repeating the same process yields (-xY(axY)m(axX)n+1)t=0 in Q*CC{X,Y}. This implies that ax=0, a contradiction. Thus for any xρ, ax and x are C-dependent. Then (a-α)ρ=0 for some αC. Replacing a with a-α, we may assume that aρ=0. Then by Lemma 2.1, d(ρ)ρ=0, contradiction.

Case 2.

Suppose that d is not Q-inner derivation. If for all xρ, d(x)xC, then [d(x),x]=0 which implies that R is commutative (see ). Therefore there exists xρ such that d(x)xC, that is, x and d(x) are linearly C-independent.

By our assumption, we have that R satisfies

[[d(xX),xX]n,[xY,d(xY)]m]t=0. By Kharchenko's Theorem , [[d(x)X+xr1,xX]n,[xY,d(x)Y+xr2]m]t=0, for all X,Y,r1,r2R. In particular for r1=r2=0, [[d(x)X,xX]n,[xY,d(x)Y]m]t=0, which is a nontrivial GPI for R, because x and d(x) are linearly C-independent, a contradiction.

We are now ready to prove our main theorem.

Proof of Theorem <xref ref-type="statement" rid="thm1.3">2.4</xref>.

Suppose that d(ρ)ρ0, then we derive a contradiction. By Lemma 2.5, R is a prime GPI ring, so is also Q by . Since Q is centrally closed over C, it follows from  that Q is a primitive ring with H=Soc(Q)0.

By our assumption and by , we may assume that [[d(x),x]n,[y,d(y)]m]t=0 is satisfied by ρQ and hence by ρH. Let e=e2ρH and yH. Then replacing x with e and y with ey(1-e) in (2.17), then right multiplying it by e, we obtain that 0=[[d(e),e]n,[ey(1-e),d(ey(1-e))]m]te=[[d(e),e]n,[ey(1-e),d(ey(1-e))]m]t-1·{[d(e),e]nj=0m(-1)j(mj)d(ey(1-e))jey(1-e)d(ey(1-e))m-je-j=0m(-1)j(mj)d(ey(1-e))jey(1-e)d(ey(1-e))m-j[d(e),e]ne}.

Now we have the fact that for any idempotent e, d(y(1-e))e=-y(1-e)d(e), ed(e)e=0 and so 0=[[d(e),e]n,[ey(1-e),d(ey(1-e))]m]t-1  ·{0-j=0m(-1)j(mj)e(-y(1-e)d(e))jy(1-e)d(ey(1-e))m-jd(e)e}. Now since for any idempotent e and for any yR, (1-e)d(ey)=(1-e)d(e)y, above relation gives 0=[[d(e),e]n,[ey(1-e),d(ey(1-e))]m]t-1·{-ej=0m(mj)(y(1-e)d(e))jy(1-e)(d(e)y(1-e))m-jd(e)e}=[[d(e),e]n,[ey(1-e),d(ey(1-e))]m]t-1{-ej=0m(mj)(y(1-e)d(e))m+1e}=[[d(e),e]n,[ey(1-e),d(ey(1-e))]m]t-1{-2me(y(1-e)d(e))m+1e}={-2me(y(1-e)d(e))m+1}te. This implies that 0=(-1)t2mt((1-e)d(e)ey)(m+1)t+1 for all yH. Since char R2, we have by Levitzki's lemma [12,Lemma 1.1] that (1-e)d(e)ey=0 for all yH. By primeness of H, (1-e)d(e)e=0. By [15,Lemma 1], since H is a regular ring, for each rρH, there exists an idempotent eρH such that r=er and erH. Hence (1-e)d(e)e=0 gives (1-e)d(e)=(1-e)d(e2)=(1-e)d(e)e=0 and so d(e)=ed(e)eHρH and d(r)=d(er)=d(e)er+ed(er)ρH. Hence for each rρH, d(r)ρH. Thus d(ρH)ρH. Set J=ρH. Then J¯=J/(JlH(J)), a prime C-algebra with the derivation d¯ such that d¯(x¯)=d(x)¯, for all xJ. By assumption, we have that [[d¯(x¯),x¯]n,[y¯,d¯(y¯)]m]t=0, for all x¯,y¯J¯. By Theorem 2.3, we have either d¯=0 or ρH¯ is commutative. Therefore we have that either d(ρH)ρH=0 or [ρH,ρH]ρH=0. Now d(ρH)ρH=0 implies that 0=d(ρρH)ρH=d(ρ)ρHρH and so d(ρ)ρ=0. [ρH,ρH]ρH=0 implies that 0=[ρρH,ρH]ρH=[ρ,ρH]ρHρH and so [ρ,ρH]ρ=0, then 0=[ρ,ρρH]ρ=[ρ,ρ]ρHρ implying that [ρ,ρ]ρ=0. Thus in all the cases we have contradiction. This completes the proof of the theorem.

3. The Case: <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M323"><mml:mrow><mml:mi>R</mml:mi></mml:mrow></mml:math></inline-formula> Semiprime Ring

In this section we extend Theorem 2.3 to the semiprime case. Let R be a semiprime ring and U be its right Utumi quotient ring. It is well known that any derivation of a semiprime ring R can be uniquely extended to a derivation of its right Utumi quotient ring U and so any derivation of R can be defined on the whole of U [7,Lemma 2].

By the standard theory of orthogonal completions for semiprime rings, we have the following lemma.

Lemma 3.1 (see [<xref ref-type="bibr" rid="B1">16</xref>, Lemma <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M331"><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula> and Theorem <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M332"><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:math></inline-formula>] or [<xref ref-type="bibr" rid="B14">7</xref>,pages 31-32]).

Let R be a 2-torsion free semiprime ring and P a maximal ideal of C. Then PU is a prime ideal of U invariant under all derivations of U. Moreover, {PUP  is  a  maximal  ideal  of  C  with  U/PU  2-torsion  free}=0.

Theorem 3.2.

Let R be a 2-torsion free semiprime ring and d a non-zero derivation of R such that [[d(x),x]n,[y,d(y)]m]t=0 for all x,yR, n,m0,t1 fixed are integers. Then d maps R into its center.

Proof.

Since any derivation d can be uniquely extended to a derivation in U, and R and U satisfy the same differential identities [7, Theorem 3], we have [[d(x),x]n,[y,d(y)]m]t=0, for all x,yU. Let P be any maximal ideal of C such that U/PU is 2-torsion free. Then by Lemma 3.1, PU is a prime ideal of U invariant under d. Set U¯=U/PU. Then derivation d canonically induces a derivation d¯ on U¯ defined by d¯(x¯)=d(x)¯ for all xU. Therefore, [[d¯(x¯),x¯]n,[y¯,d¯(y¯)]m]t=0, for all x¯,y¯U¯. By Theorem 2.3, either d¯=0 or [U¯,U¯]=0, that is, d(U)PU or [U,U]PU. In any case d(U)[U,U]PU for any maximal ideal P of C. By Lemma 3.1, {PUP  is  a  maximal  ideal  of  C  with  U/PU  2-torsion  free}=0. Thus d(U)[U,U]=0. Without loss of generality, we have d(R)[R,R]=0. This implies that 0=d(R2)[R,R]=d(R)R[R,R]+Rd(R)[R,R]=d(R)R[R,R]. Therefore [R,d(R)]R[R,d(R)]=0. By semiprimeness of R, we have [R,d(R)]=0, that is, d(R)Z(R). This completes the proof of the theorem.

PosnerE. C.Derivations in prime ringsProceedings of the American Mathematical Society1957810931100MR009586310.2307/2032686LanskiC.Differential identities, Lie ideals, and Posner's theoremsPacific Journal of Mathematics19881342275297MR961236ZBL0614.16028BellH. E.MartindaleW. S.IIICentralizing mappings of semiprime ringsCanadian Mathematical Bulletin198730192101MR879877ZBL0614.16026HersteinI. N.A note on derivationsCanadian Mathematical Bulletin1978213369370MR0506447ZBL0412.16018De FilippisV.On derivations and commutativity in prime ringsInternational Journal of Mathematics and Mathematical Sciences200469–723859386510.1155/S0161171204403536MR2129413ZBL1078.16036BeidarK. I.MartindaleW. S.IIIMikhalevA. V.Rings with Generalized Identities1996196New York, NY, USAMarcel Dekkerxiv+522Monographs and Textbooks in Pure and Applied MathematicsMR1368853LeeT. K.Semiprime rings with differential identitiesBulletin of the Institute of Mathematics Academia Sinica19922012738MR1166215ZBL0769.16017BrešarM.One-sided ideals and derivations of prime ringsProceedings of the American Mathematical Society19941224979983MR120548310.2307/2161163ZBL0820.16032KharchenkoV. K.Differential identities of prime ringsAlgebra i Logika1978172155168MR541758LanskiC.An Engel condition with derivationProceedings of the American Mathematical Society19931183731734MR113285110.2307/2160113ZBL0821.16037MartindaleW. S.IIIPrime rings satisfying a generalized polynomial identityJournal of Algebra196912576584MR023889710.1016/0021-8693(69)90029-5ZBL0175.03102HersteinI. N.Topics in Ring Theory1969, Chicago, Ill, USAThe University of Chicago Pressxi+132MR0271135BellH. E.DengQ.On derivations and commutativity in semiprime ringsCommunications in Algebra1995231037053713MR134825910.1080/00927879508825427ZBL0832.16033ChuangC.-L.GPI's having coefficients in Utumi quotient ringsProceedings of the American Mathematical Society19881033723728MR94764610.2307/2046841ZBL0656.16006FaithC.UtumiY.On a new proof of Litoff's theoremActa Mathematica Academiae Scientiarum Hungaricae196314369371MR015585810.1007/BF01895723ZBL0147.28602BeidarK. I.Rings of quotients of semiprime ringsVestnik Moskovskogo Universiteta19783353643MR516019ZBL0413.16004