Vanishing Power Values of Commutators with Derivations on Prime Rings

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Volume 2009, Article ID 582181,8pages doi:10.1155/2009/582181

Research Article

Vanishing Power Values of Commutators with

Derivations on Prime Rings

Basudeb Dhara

Department of Mathematics, Belda College, Belda, Paschim Medinipur 721424, India

Correspondence should be addressed to Basudeb Dhara,basu dhara@yahoo.com

Received 30 August 2009; Accepted 14 December 2009

Recommended by Howard Bell

LetRbe a prime ring of charR /2,da nonzero derivation ofRandρa nonzero right ideal ofR

such thatdx, xn,y, dym t

0 for allx, yρ, wheren≥0,m≥0,t≥1 are fixed integers. If ρ, ρρ /0, thendρρ0.

Copyrightq2009 Basudeb Dhara. This 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.

1. Introduction

Throughout this paper, unless specifically stated,Ralways denotes a prime ring with center ZRand extended centroidC,Qthe Martindale quotients ring. Letnbe a positive integer. For given a, bR, let a, b0 a and let a, b1 be the usual commutator abba, and inductively forn >1,a, bn a, bn−1, b. Bydwe mean a nonzero derivation inR.

A well-known result proven by Posner1states that ifdx, x, y 0 for allx, yR, then R is commutative. In2, Lanski generalized this result of Posner to the Lie ideal. Lanski proved that ifUis a noncommutative Lie ideal ofRsuch thatdx, x, y 0 for allxU, yR, then eitherRis commutative or charR 2 andRsatisfiesS4, the standard identity in four variables. Bell and Martindale III3studied this identity for a semiprime ringR. They proved that ifRis a semiprime ring anddx, x, y 0 for allxin a non-zero left ideal ofRandyR, thenRcontains a non-zero central ideal. Clearly, this result says that ifRis a prime ring, thenRmust be commutative.

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In continuation of these previous results, it is natural to consider the situation when dx, xn,y, dymt0 for allx, yρ,n, m≥0, t≥1 are fixed integers. We have studied this identity in the present paper.

It is well known that any derivation of a prime ringRcan be uniquely extended to a derivation ofQ, and so any derivation ofRcan be defined on the whole ofQ. MoreoverQis a prime ring as well asRand the extended centroidCofRcoincides with the center ofQ. We refer to6,7for more details.

Denote byQCC{X, Y}the free product of theC-algebraQandC{X, Y}, the free

C-algebra in noncommuting indeterminatesX, Y.

2. The Case:

R

Prime Ring

We need the following lemma.

Lemma 2.1. Letρbe a non-zero right ideal ofRandda derivation ofR. Then the following conditions are equivalent:id is an inner derivation induced by somebQsuch thatbρ 0;iidρρ0

(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 9. 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 fr1, r2, . . . , rn, dr1, dr2, . . . , drn

0 for anyr1, r2, . . . , rnI,then eitheriIsatisfies the generalized polynomial identity

fr1, r2, . . . , rn, x1, x2, . . . , xn 0, 2.1

oriidisQ-inner, that is, for someqQ, dx q, xandI satisfies the generalized polynomial identity

fr1, r2, . . . , rn,

q, r1

,q, r2

, . . . ,q, rn

0. 2.2

Theorem 2.3. Let R be a prime ring of char R /2 and d a derivation of R such that

dx, xn,y, dymt 0 for all x, yR, where n ≥ 0, m ≥ 0, t ≥ 1 are fixed integers.

ThenRis commutative ord0.

Proof. LetRbe noncommutative. Ifdis notQ-inner, then by Kharchenko’s Theorem9

gx, y, u, v u, xn,y, vmt0, 2.3

for allx, y, u, vR. This is a polynomial identity and hence there exists a fieldF such that RMkFwithk >1,andRandMkFsatisfy the same polynomial identity10, Lemma

1. But by choosingue12, xe11, ve11andye21, we get

0u, xn,y, vmt −1tne11 −te22

, 2.4

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Now, letdbeQ-inner derivation, saydadafor someaQ, that is,dx a, x for allxR, then we have

a, xn1,y,a, ymt0, 2.5

for allx, yR. Sinced /0,a /Cand henceRsatisfies a nontrivial generalized polynomial identityGPI. By11, it follows thatRCis a primitive ring withHSocRC/0,andeHe is finite dimensional overCfor any minimal idempotenteRC. Moreover we may assume thatHis noncommutative; otherwise,Rmust be commutative which is a contradiction.

Notice thatHsatisfiesa, xn1,y,a, ymt 0see10, Proof of Theorem 1. For any idempotenteHandxH,we have

0 a, en1,ex1−e,a, ex1−emt. 2.6

Right multiplying bye, we get

0 a, en1,ex1−e,a, ex1−emte

a, en1,ex1−e,a, ex1−emt−1

· {a, en1ex1−e,a, ex1−emeex1−e,a, ex1−ema, en1e}

a, en1,ex1−e,a, ex1−emt−1

·

⎧ ⎨

a, en1 ⎛ ⎝m

j0 −1j

m

j

a, ex1−ejex1−ea, ex1−emj ⎞ ⎠e

⎛ ⎝m

j0 −1j

m

j

a, ex1−ejex1−ea, ex1−emj

a, en1e ⎫ ⎬ ⎭

a, en1,ex1−e,a, ex1−emt−1

·

⎧ ⎨ ⎩0−

⎛ ⎝m

j0 −1j

m

j

ex1−eajex1−eaex1−emj ⎞ ⎠ae

⎫ ⎬ ⎭

a, en1,ex1−e,a, ex1−emt−1 ⎛ ⎝m

j0

m

j

ex1−eam1 ⎞ ⎠e

−2ma, en1,ex1−e,a, ex1−emt−1ex1−eam1e

t2mtex1eam1te.

2.7

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Soacommutes with all idempotents inH. SinceHis a simple ring, either His generated by its idempotents orH does not contain any nontrivial idempotents. The first case gives aCcontradictingd /0. In the last case,His a finite dimensional division algebra overC. This implies thatH RC Q andaH. By 10, Lemma 2, there exists a field F such thatHMkFandMkFsatisfiesa, xn1,y,a, ym

t. Then by the same argument as

earlier,acommutes with all idempotents inMkF, again giving the contradictionaC, that

is,d0. This completes the proof of the theorem.

Theorem 2.4. LetRbe a prime ring of charR /2,da non-zero derivation ofRandρa non-zero right ideal ofRsuch thatdx, xn,y, dymt0 for allx, yρ, wheren≥0, m≥0, t≥1 are fixed

integers. Ifρ, ρρ /0, thendρρ0.

We begin the proof by proving the following lemma.

Lemma 2.5. Ifdρρ /0 anddx, xn,y, dymt0 for allx, yρ, m, n≥0, t≥1 are fixed

integers, thenRsatisfies nontrivial generalized polynomial identity (GPI).

Proof. Suppose on the contrary thatRdoes not satisfy any nontrivial GPI. We may assume thatRis noncommutative; otherwise,Rsatisfies trivially a nontrivial GPI. We consider two cases.

Case 1. Suppose that disQ-inner derivation induced by an elementaQ. Then for any xρ,

a, xXn1,xY,a, xYmt 2.8

is a GPI forR, so it is the zero element inQCC{X, Y}. Expanding this, we get

a, xXn1m

j0 −1j

m

j

a, xYjxYa, xYmj

m

j0 −1j

m

j

a, xYjxYa, xYmja, xXn1

AX, Y 0,

2.9

whereAX, Y a, xXn1,xY,a, xYmt−1. Ifax andx are linearlyC-independent for somexρ,then

axXn1m

j0 −1j

m

j

a, xYjxYa, xYmj

m

j0 −1j

m

j

axYjxYa, xYmja, xXn1

AX, Y 0.

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Again, sinceaxandxare linearlyC-independent, above relation implies that

xYa, xYma, xXn1AX, Y 0, 2.11

and so

xYaxYmaxXn1AX, Y 0. 2.12

Repeating the same process yields

xYaxYmaxXn1t0 2.13

inQCC{X, Y}. This implies thatax 0, a contradiction. Thus for anyxρ,axandxare

C-dependent. Thenaαρ0 for someαC. Replacingawithaα, we may assume that 0. Then by Lemma2.1,dρρ0, contradiction.

Case 2. Suppose thatdis notQ-inner derivation. If for allxρ,dxxC, thendx, x 0 which implies thatR is commutative see13. Therefore there exists xρ such that dx/xC, that is,xanddxare linearlyC-independent.

By our assumption, we have thatRsatisfies

dxX, xXn,xY, dxYmt0. 2.14

By Kharchenko’s Theorem9,

dxXxr1, xXn,xY, dxYxr2mt0, 2.15

for allX, Y, r1, r2∈R. In particular forr1r2 0,

dxX, xXn,xY, dxYmt0, 2.16

which is a nontrivial GPI for R, because x and dx are linearly C-independent, a contradiction.

We are now ready to prove our main theorem.

Proof of Theorem2.4. Suppose thatdρρ /0,then we derive a contradiction. By Lemma2.5,R is a prime GPI ring, so is alsoQby14. SinceQis centrally closed overC, it follows from 11thatQis a primitive ring withHSocQ/0.

By our assumption and by7, we may assume that

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is satisfied byρQand hence byρH. Letee2ρHandyH. Then replacingxwitheand ywithey1−ein2.17, then right multiplying it bye,we obtain that

0de, en,ey1−e, dey1−emte

de, en,ey1−e, dey1−emt−1

·

⎧ ⎨

de, en m

j0 −1j

m

j

dey1−ejey1−edey1−emje

m

j0 −1j

m

j

dey1−ejey1−edey1−emjde, ene ⎫ ⎬ ⎭.

2.18

Now we have the fact that for any idempotente,dy1−eey1−ede,edee0 and so

0de, en,ey1−e, dey1−emt−1

·

⎧ ⎨ ⎩0−

m

j0 −1j

m

j

ey1−edejy1−edey1−emjdee ⎫ ⎬ ⎭.

2.19

Now since for any idempotenteand for anyyR,1−edey 1−edey, above relation gives

0de, en,ey1−e, dey1−emt−1

·

⎧ ⎨ ⎩−e

m

j0

m

j

y1−edejy1−edey1−emjdee ⎫ ⎬ ⎭

de, en,ey1−e, dey1−emt−1 ⎧ ⎨ ⎩−e

m

j0

m

j

y1−edem1e ⎫ ⎬ ⎭

de, en,ey1−e, dey1−emt−1−2mey1−edem1e

−2mey1−edem1te.

2.20

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ThenJ J/JlHJ,a primeC-algebra with the derivationdsuch thatdx dx, for all

xJ. By assumption, we have that

dx, x

n,

y, dy

m

t

0, 2.21

for all x, yJ. By Theorem 2.3, we have either d 0 or ρH is commutative. Therefore we have that either dρHρH 0 or ρH, ρHρH 0. Now dρHρH 0 implies that 0 dρρHρH dρρHρH and so dρρ 0. ρH, ρHρH 0 implies that 0 ρρH, ρHρH ρ, ρHρHρH and soρ, ρHρ 0,then 0 ρ, ρρHρ ρ, ρρHρ implying thatρ, ρρ 0. Thus in all the cases we have contradiction. This completes the proof of the theorem.

3. The Case:

R

Semiprime Ring

In this section we extend Theorem2.3to the semiprime case. LetRbe a semiprime ring and Ube 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 ringUand so any derivation ofRcan be defined on the whole ofU7, Lemma 2.

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

Lemma 3.1see16, Lemma 1 and Theorem 1or7, pages 31-32. LetRbe a 2-torsion free semiprime ring and P a maximal ideal of C. Then P U is a prime ideal of U invariant under all derivations ofU. Moreover,{P U|P is a maximal ideal ofCwithU/P U 2-torsion free}0.

Theorem 3.2. LetRbe a 2-torsion free semiprime ring and da non-zero derivation ofRsuch that

dx, xn,y, dymt0 for allx, yR,n, m≥0, t≥1 fixed are integers. ThendmapsRinto its center.

Proof. Since any derivationdcan be uniquely extended to a derivation inU,and Rand U satisfy the same differential identities7, Theorem 3, we have

dx, xn,y, dymt0, 3.1

for allx, yU. LetPbe any maximal ideal ofCsuch thatU/P U is 2-torsion free. Then by Lemma3.1,P Uis a prime ideal ofUinvariant underd. SetU U/P U.Then derivationd canonically induces a derivationdonUdefined bydx dxfor allxU. Therefore,

dx, x

n,

y, dy

m

t

0, 3.2

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{P U | P is a maximal ideal of CwithU/P U 2-torsion free} 0. ThusdUU, U 0. Without loss of generality, we havedRR, R 0. This implies that

0dR2R, R dRRR, R RdRR, R dRRR, R. 3.3

ThereforeR, dRRR, dR 0. By semiprimeness of R, we have R, dR 0, that is, dRZR. This completes the proof of the theorem.

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