Generalized Derivations in Semiprime Gamma Rings

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Volume 2012, Article ID 270132,14pages doi:10.1155/2012/270132

Research Article

Generalized Derivations in

Semiprime Gamma Rings

Kalyan Kumar Dey,

1

Akhil Chandra Paul,

1

and Isamiddin S. Rakhimov

2

1Department of Mathematics, University of Rajshahi, Rajshahi 6205, Bangladesh

2Department of Mathematics, FS, and Institute for Mathematical Research (INSPEM),

Universiti Putra Malaysia, 43400 Serdang, Malaysia

Correspondence should be addressed to Kalyan Kumar Dey,kkdmath@yahoo.com

Received 10 November 2011; Revised 5 December 2011; Accepted 5 December 2011

Academic Editor: Christian Corda

Copyrightq2012 Kalyan Kumar Dey et al. 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.

Let M be a 2-torsion-free semiprimeΓ-ring satisfying the conditionaαbβcaβbαcfor alla, b, c

M, α, β∈Γ, and letD:MMbe an additive mapping such thatDxαx Dxαxxαdxfor

allxM, α∈Γand for some derivation d of M. We prove that D is a generalized derivation.

1. Introduction

Hvala1first introduced the generalized derivations in rings and obtained some remarkable results in classical rings. Daif and Tammam El-Sayiad2studied the generalized derivations in semiprime rings. The authors consider an additive mappingG :RRof a ringRwith the propertyGx2 GxxxDxfor some derivationDofR. They prove thatGis a Jordan

generalized derivation.

Aydin3studied generalized derivations of prime rings. The author proved that ifIis an ideal of a noncommutative prime ringR,ais a fixed element ofRandFis an generalized derivation onRassociated with a derivationdthen the conditionFx, a 0 orFx, a 0 for allxIimpliesdx λx, a.

C¸ even and ¨Ozt ¨urk4have dealt with Jordan generalized derivations inΓ-rings and they proved that every Jordan generalized derivation on some Γ-rings is a generalized derivation.

Generalized derivations of semiprime rings have been treated by Ali and Chaudhry

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Rassociated with a derivationdthendxy, z 0 for allx, y, zRanddis central. They characterized a decomposition ofRrelative to the generalized derivations.

Atteya6proved that if Uis nonzero ideal of a semiprime ringR and R admits a generalized derivationD such thatDxyxyZRthen Rcontains a nonzero central ideal.

Rehman7,8studied the commutativity of a ringRby means of generalized deriva-tions acting as homomorphisms and antihomomorphisms.

In this paper, we prove the following results

LetMbe a 2-torsion-free semiprimeΓ-ring satisfying the following assumption:

aαbβcaβbαc ∀a, b, c∈M, α, β∈Γ, ∗

and D : MM be an additive mapping. If there exists a derivationdof M such that

Dxαx Dxαxxαdxfor allxM,α∈Γ, thenDis a Jordan generalized derivation.

2. Preliminaries

LetMandΓbe additive abelian groups.Mis called aΓ-ring if there exists a mappingM× M×MMsuch that for alla, b, cM,α, β∈Γthe following conditions are satisfied:

iaβbM,

ii abαcaαcbαc,aαβbaαbaβb,aαbc aαbaαc,aαbβcaαbβc.

For anya, bMand forα, β∈Γthe expressionsaαb−bαais denoted bya, bαandαaβ−βaα

are denoted byα, βa. Then one has the following identities:

aβb, cαaβb, cα a, cαβba

β, αcb,

a, bβcαbβa, cα a, bαβcb

β, αac, 2.1

for alla, b, cMand for allα, β∈Γ. Using the assumption∗the above identities reduce to

aβb, cαaβb, cα a, cαβb,

a, bβcαbβa, cα a, bαβc,

2.2

for alla, b, cMandα,β∈Γ.

FurtherMstands for a primeΓ-ring with centerZM. The ringMisn-torsion-free if nx 0,xMimplies x 0, wherenis a positive integer,Mis prime if aΓMΓb 0 impliesa0 orb0, and it is semiprime ifaΓMΓa0 impliesa0. An additive mapping

T : MMis called a leftrightcentralizer ifTxαy TxαyTxαy xαTyfor

x, yM,α∈Γand it is called a Jordan leftrightcentralizer if

Txαx TxαxTxαx xαTx ∀x∈M, α∈Γ. 2.3

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x, yM,α ∈ Γand it is called a Jordan derivation if Dxαx DxαxxαDxfor all

xM,α ∈ Γ. A derivationD is inner if there existsaM, such thatDx aαxxαa

holds for allxM,α∈Γ. Every derivation is a Jordan derivation. The converse is in general not true. An additive mappingD : MMis said to be a generalized derivation if there exists a derivationd : MMsuch that Dxαy Dxαyxαdyfor allx, yM,

α ∈ Γ. The maps of the formxaαxxαbwherea,bare fixed elements inMand for all α ∈ Γ called the generalized inner derivation. An additive mappingD : MM is said to be a Jordan generalized derivation if there exists a derivation d : MM such

thatDxαx Dxαxxαdxfor allxM,α ∈ Γ. Hence the concept of a generalized

derivation covers both the concepts of a derivation and a left centralizers and the concept of a Jordan generalized derivation covers both the concepts of a Jordan derivation and a left Jordan centralizers. An example of a generalized derivation and a Jordan generalized derivation is given in4.

3. Main Results

We start from the following subsidiary results.

Lemma 3.1. LetMbe a semiprimeΓ-ring. Ifa, bMare such thataαxβb 0 for allxM,

α, β∈Γ, thenaαbbαa0.

Proof. LetxM. Then

aαbβxγaαbaαbβxγaαb0,

bαaβxγbαabαaβxγbαa0. 3.1

By semiprimeness ofMwith respect toβ, γ ∈Γ, it follows thataαbbαa0.

Lemma 3.2. Let M be a semiprime Γ-ring and θ, ϕ : M× MM biadditive mappings. If

θx, yαwβϕx, y 0 for allx, y, wM, thenθx, yαwβϕu, v 0 for allx, y, u, v, wM,

α, β∈Γ.

Proof. First we replacexwithxuin the relationθx, yαwβϕx, y 0, and use the

biaddi-tivity of theθandϕ. Then we have

θxu, yαwβϕxu, y0⇒θx, yαwβϕu, y−θu, yαwβϕx, y. 3.2

Then

θx, yαwβϕu, yγzδθx, yαwβϕu, y

θu, yαwβϕx, yγzδθu, yαwβϕu, y0. 3.3

Henceθx, yαwβϕu, y 0 by semiprimeness ofMwith respect toγ, δ∈Γ.

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Lemma 3.3. LetMbe a semiprimeΓ-ring satisfying the assumption∗andabe a fixed element of

M. Ifaβx, yα0 for allx, yM,α, β∈Γ, thenaZM.

Proof. First we calculate the following expressions using the assumption∗,

z, aαβxδz, aαzαaβxδz, aαaαzβxδz, aα

zαaβz, xαaδzαaβz, xδαaaαz, zδxαaβaαz, zδxβαa.

3.4

Since aβx, yα 0 for all x, yM, α, β ∈ Γ, we get z, aαβxδz, aα 0. By the semiprimeness ofMwe getz, aα0 for allα∈Γ. HenceaZM.

Lemma 3.4. LetMbe aΓ-ring satisfying the condition∗andD:MMbe a Jordan generalized

derivation with the associated derivationd. Letx, y, zMandα, β∈Γ. Then

iDxαyyαx DxαyDyαxxαdy yαdx.

In particular, ifMis 2-torsion-free, then

iiDxαyβx Dxαyβxxβdyβx,

iiiDxαyβzzβyαx DxαyβzDzβyαxxαdyβz zβdyαx.

Proof. iWe haveDxαx Dxαxxαdx, for allxM,α∈Γ. Then replacingxbyxy,

and following the series of implications below we get the result:

Dxyαxy

Dxyαxyxyαdxy

Dxαxxαyyαxyαy

DxαxDyαxDxαyDyαxxαdyxαdx yαdyyαdx

DxαxyαyDxαyyαx

Dxαxxαdx DyαyyαdyDxαy

Dyαxxαdyyαdx

Dxαyyαx

DxαyDyαxxαdyyαdx, ∀x, y∈M, α∈Γ.

3.5

iiReplaceybyxβyyβxin the above relation3.5, then we get,

Dxαxβyyβxxβyyβxαx

DxαxβyyβxDxβyyβxαx

xαdxβyyβxxβyyβxαdx, ∀x, y, z∈M, α, β∈Γ.

DxαxβyyβxαxDxαyβxDxβyαx

DxαxβyDxαyβxDxβyαxDyβxαxxβdyαxyβdxαx

xαdxβyyβxxβyyβxαdx, ∀x, y, z∈M, α, β∈Γ.

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Using the assumption∗, we conclude that

Dxαxβyyβxαx2DxαyβxDxαxβyDxαyβxDxβyαxDyβxαx

xβdyαxyβdxαxxαdxβyyβx

xβyyβxαdx, ∀x, y, z∈M, α, β∈Γ.

3.7

Again, replacingxbyxβxin3.5

DxβxαyyαxβxDxβxαyDyαxβxxβxαdy

yαdxβx, ∀x, y∈M, α, β∈Γ,

DxβxαyyαxβxDxβxαyxβdxαyDyαxβxxβxαdy

yαdxβxyαxβdx, ∀x, y∈M, α, β∈Γ.

3.8

Adding both sides 2Dxαyβx, we get,

Dxβxαyyαxβx2Dxαyβx

DxβxαyxβdxαyDyαxβxxβxαdy

yαdxβxyαxβdx 2Dxαyβx, ∀x, y∈M, α, β∈Γ.

3.9

Comparing3.7and3.9we obtain,

2Dxαyβx2Dxαyβxxαyβdx xαdyβx. 3.10

SinceMis 2-torsion-free, it gives

DxαyβxDxαyβxxαdyβx. 3.11

iiiReplacexforxzin3.12, we get,

Dxzαyβxz

Dxzαyβxz xzαdyβxz

Dxαyβxxαyβzzαyβxzαyβz

DxαyβxDzαyβxDxαyβzDzαyβzxαdyβx

xαdyβzzαdyβxyβz

DxαyβxzαyβzDxαyβzzαyβx

DxαyβxDzαyβzxαdyβxzαdyβz

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Dxαyβzzαyβx

DxαyβzDzαyβxxαdyβzzαdyβx

Dxαyβzzβyαx

DxαyβzDzαyβxxαdyβzzαdyβx, ∀x, y∈M, α, β∈Γ.

3.12

Definition 3.5. LetMbe aΓ-ring andD :MMbe a Jordan generalized derivation with

the associated derivationd. DefineGαx, y DxαyDxαyxαdyfor allx, yM

andα∈Γ.

Lemma 3.6. The functionGαx, yhas the following properties:

iGαx, y Gαy, x 0.

iiGαx, yz Gαx, y Gαx, z.

iiiGαxy, z Gαx, z Gαy, z.

ivGαβx, y Gαx, y Gβx, y.

Proof. The results easily follow fromLemma 3.4i.

Remark 3.7. Dis a generalized derivation if and only ifGαx, y 0 for allx, yM,α∈Γ.

Theorem 3.8. LetMbe a 2-torsion-free semiprimeΓ-ring satisfying the condition∗. Letx, y, z

Mandβ, δ∈Γ. Then

iGαx, yβzδx, yα0 for allzM,

iiGαx, yβx, yα0.

Proof. iFromLemma 3.4iiiwe get

DxαyβzzβyαxDxαyβzDzβyαxxαdyβzzβdyαx. 3.13

We set

Axαyβzδyαxyαxβzδxαy. 3.14

Since

DA Dxαyβzδyαxyαxβzδxαy

Dxαyβzδyαxyαxβzδxαy

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DxαyβzδyαxxαdyβzδyαxxαyβdzδyαxDyαxβzδxαy

yαdxβzδxαyyαxβdzδxαy, ∀x, y, z∈M, α, β, δ∈Γ.

3.15

Again

DA Dxαyβzδyαxyαxβzδxαy

Dxαyβzδyαxyαxβzδxαy

DxαyβzδyαxDyαxβzδxαyxαyβdzδyαx

yαxβdzδxαy, ∀x, y, z∈M, α, β, δ∈Γ.

3.16

From3.15and3.16we find,

DxαyDxαyxαdyβzδyαxDyαxDyαxyαdxβzδxαy0.

Gαx, yβzδyαxGαy, xβzδxαy0,

Gαx, yβzδyαxGαx, yβzδxαy0,

Gαx, yβzδy, xα0,

Gαx, yβzδx, yα0, forx, yM, α, β, δ∈Γ.

3.17

iiAccording toLemma 3.4iiwe have,

DxαyβzDxαyβzxαdyβz. 3.18

Replacezbyxαyin3.18we find

DxαyβxαyDxαyβxαyxαdyβxαy

DxαyβxαyDxαyβxαyxαdyβxαy

DxαyβxαyxαyβdxαyDxαyβxαyxαdyβxαy0

DxαyDxαyxαdyβxαyxαyβdxαyxαyβdxαy0

Gαx, yβxαy0.

3.19

SimilarlyGαx, yβyαx0. Therefore,

Gαx, yβxαyGαx, yβyαxGαx, yβx, yα0. 3.20

Lemma 3.9. Gαx, yZM, for allx, yM,α∈Γ.

Proof. FromTheorem 3.8ii, we have Gαx, yβx, yα 0. Therefore due toLemma 3.4ii

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Theorem 3.10. Let M be a 2-torsion-free semiprime Γ-ring satisfying the assumption ∗ and

D : MMbe a Jordan generalized derivation with associated derivationdonM. Then Dis a

generalized derivation.

Proof. In particular,rγGαx, y,Gαx, yγrZMfor allrM,α, γ ∈Γ. This gives

xαGαx, yδGαx, yβyGαx, yδGαx, yβyαx

yβGαx, yδGαx, yαx

yαGαx, yδGαx, yβx.

3.21

Then 4DxαGαx, yδGαx, yβy 4DyβGαx, yδGαx, yαx. Now we will compute each side of this equality by using3.15and the above relation,

4DxαGαx, yδGαx, yβy

2DxαGαx, yδGαx, yβyGαx, yδGαx, yβyαx

2DxαGαx, yδGαx, yβy2xαdGαx, yδGαx, yβy

2DGαx, yδGαx, yβyαx2Gαx, yδGαx, yβyαdx

2DxαGαx, yδGαx, yβyDGαx, yδGαx, yβyyβGαx, yδGαx, yαx

2xαdGαx, yδGαx, yβy2Gαx, yδGαx, yβyαdx

2DxαGαx, yδGαx, yβyDGαx, yδGαx, yβyαx

Gαx, yδdGαx, yβyαxDyβGαx, yδGαx, yαx

Gαx, yδGαx, yβdyαxyβdGαx, yδGαx, yαx

2xαdGαx, yδGαx, yβy2Gαx, yδGαx, yβyαdx.

3.22

So we get

4DxβGαx, yδGαx, yαy

2DxβGαx, yδGαx, yαyDGαx, yδGαx, yβyαx

Gαx, yδdGαx, yβyxDyβGαx, yδGαx, yαx

Gαx, yδGαx, yβdyαxyβdGαx, yδGαx, yαx

2xαdGαx, yδGαx, yβy2Gαx, yδGαx, yβyαdx, x, yM, α, β, δ∈Γ.

3.23

Moreover,

4DyβGαx, yδGαx, yαx

2DyβGαx, yδGαx, yαxGαx, yδGαx, yβxαy

2DyβGαx, yδGαx, yαx2yβdGαx, yδGαx, yαx

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2DyβGαx, yδGαx, yαxDGαx, yδGαx, yβxxβGαx, yδGαx, yαy

2yβdGαx, yδGαx, yαx2Gαx, yδGαx, yβxαdy

2DyβGαx, yδGαx, yαxDGαx, yδGαx, yβxαy

Gαx, yδdGαx, yβxαyDxβGαx, yδGαx, yαy

Gαx, yδGαx, yαdxβyxαdGαx, yδGαx, yβy

2yβdGαx, yδGαx, yαx2Gαx, yδGαx, yαxβdy.

3.24

So we get

4DyβGαx, yδGαx, yαx

2DyβGαx, yδGαx, yαxDGαx, yδGαx, yαxβy

Gαx, yδdGαx, yαxβyDxβGαx, yδGαx, yαy

Gαx, yδGαx, yαdxβyxαdGαx, yδGαx, yβy

2yβdGαx, yδGαx, yαx2Gαx, yδGαx, yαxβdy,

x, yM, α, β, δ∈Γ.

3.25

Comparing the results of3.22and3.24and using the above relations

Gαx, yβyαxGαx, yβyαx

xαGαx, yβy

xαGαx, yβy

Gαx, yβxαy,

Gαx, yδdGαx, yβyαxdGαx, yδGαx, yβyαx

dGαx, yδGαx, yαxβy

Gαx, yδdGαx, yαxβy,

xβGαx, yδdGαx, yαydGαx, yαxδGαx, yβy

dGαx, yδGαx, yβxαy

dGαx, yδGαx, yαyβx

Gαx, yβyδdGαx, yαx

yβGαx, yδdGαx, yαx,

3.26

we obtain

DxαGαx, yδGαx, yβyxαGαx, yδGαx, yβdy

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which gives

ϕαx, yβGαx, yδGαx, yϕαy, xαGαx, yδGαx, y, 3.28

whereϕαx, ystands forDxαyxαdy. On the other hand, we have

4DxαyβGαx, yδGαx, y4DxαGαx, yδyβGαx, y. 3.29

Now we use3.15and the properties ofGαx, y, to derive

4DxαyβGαx, yδGαx, y

2DxαyβGαx, yδGαx, yGαx, yδGαx, yαxβy

2DxαyβGαx, yδGαx, y2xαyβdGαx, yδGαx, y

2DGαx, yδGαx, yαxβy2Gαx, yδGαx, yβdxαy,

3.30

which gives

4DxαyβGαx, yδGαx, y2DxαyβGαx, yδGαx, y2xαyβdGαx, yδGαx, y

2DGαx, yδGαx, yαxβy

2Gαx, yδGαx, yβdxαy, x, yM, α, β, δ∈Γ.

3.31

Moreover,

4DxαGαx, yδyβGαx, y

2DxαGαx, yδyβGαx, yyβGαx, yδxαGαx, y

2DGαx, yαxδGαx, yβy2Gαx, yαxδdGαx, yβy

2DGαx, yβyδGαx, yαx2Gαx, yβyδdGαx, yαx

DxαGαx, yGαx, yαxδGαx, yβy2Gαx, yαxδdGαx, yβy

DyβGαx, yGαx, yβyδGαx, yαx2Gαx, yβyδdGαx, yαx

DxαGαx, yδGαx, yβyDGαx, yδGαx, yαxβy

xαdGαx, yδGαx, yβyGαx, yαdxδGαx, yβy

2Gαx, yαxδdGαx, yβyDyβGαx, yδGαx, yαx

DGαx, yδGαx, yβyαxyβdGαx, yδGαx, yαx

Gαx, yβdyδGαx, yαx2Gαx, yβyδdGαx, yαx.

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So we obtain

4DxαGαx, yδyβGαx, y

DxαGαx, yδGαx, yβyDGαx, yδGαx, yβxαy

xαdGαx, yδGαx, yβyGαx, yαdxδGαx, yβy

2Gαx, yαxδdGαx, yβyDyβGαx, yδGαx, yαx

DGαx, yδGαx, yβyαxyβdGαx, yδGαx, yαx

Gαx, yβdyδGαx, yαx2Gαx, yβyδdGαx, yαx,

x, yM, α, β, δ∈Γ.

3.33

Comparing3.31and3.33, we derive

2DxαyβGαx, yδGαx, y

ϕαx, yβGαx, yδGαx, yϕαy, xβGαx, yδGαx, y, x, yM, α, β, δ∈Γ.

3.34

Finally using 3.31 we get DxαyβGαx, yδGαx, y ϕαx, yβGαx, yδGαx, y. But

Gαx, y Dxαyϕαx, y. By the semiprimeness ofM, we haveϕαx, yβGαx, y 0.

Again by the primeness ofM, we getGαx, y. The proof is complete.

It is clear that if we let the derivationdto be the zero derivation in the above theorem, we get the following result.

Theorem 3.11. Let M be a 2-torsion-free semiprime Γ-ring and D : MM be an additive

mapping which satisfiesDxαx Dxαxfor allxM,α∈Γ. ThenDis a left centralizer

Proof. We have

Dxαx Dxαx. 3.35

If we replacexbyxy, we get

DxαyyαxDxαyDyαx. 3.36

By replacingywithxαyyαxand using3.5, we arrive at

DxαxαyyαxxαyyαxαxDxαxαyDxαyαxDxαyαxDyαxαx.

3.37

But on the other hand,

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Comparing3.37and3.38we obtain

DxαyαxDxαyαx. 3.39

If we linearize3.39inx, we get

DxαyαzzαyαxDxαyαzDzαyαx. 3.40

Now we shall computeDxαyαzαyαxyαxαzαxαyin two different ways. If we use3.39 we have

DxαyαzαyαxyαxαzαxαyDxαyαzαyαxDyαxαzαxαy. 3.41

But if we use3.40we have

DxαyαzαyαxyαxαzαxαyDxαyαzαyαxDyαxαzαxαy. 3.42

Comparing3.41and 3.42and introducing a bi-additive mappingGαx, y Dxαy

Dxαywe arrive at

Gαx, yαzαyαxGαy, xαzαxαy0. 3.43

Equality3.36can be rewritten in this notation asGαx, y −Gαy, x. Using this fact and

3.43we obtain

Gαx, yαzαx, yα0. 3.44

Using firstLemma 3.2and thenLemma 3.1we have

Gαx, yαzαu, vα0. 3.45

Now fix somex, yMand usingLemma 3.3we getGαx, yZ. In particular,rβGαx, y,Gαx, yβrZfor allrM. This gives

xβGαx, yδGαx, yβyGαx, yΓGαx, yΓyΓx

yΓGαx, yΓGαx, yΓx

yΓGαx, yΓGαx, yΓx.

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Therefore 4DxΓGαx, yΓGαx, yΓy 4DyΓGαx, yΓGαx, yΓx. Both sides of this equality will be computed in few steps using3.36,

2DxΓGαx, yΓGαx, yΓyGαx, yΓGαx, yΓyΓx

2DyΓGαx, yΓGαx, yΓxGαx, yΓGαx, yΓxΓy,

2DxΓGαx, yΓGαx, yΓy2DGαx, yΓGαx, yΓyΓx

2DyΓGαx, yΓGαx, yΓx2DGαx, yΓGαx, yΓxΓy,

2DxΓGαx, yΓGαx, yΓyDGαx, yΓGαx, yΓyyΓGαx, yΓGαx, yΓx

2DyΓGαx, yΓGαx, yΓxDGαx, yΓGαx, yΓxxΓGαx, yΓGαx, yΓy,

2DxΓGαx, yΓGαx, yΓyDGαx, yΓGαx, yΓyΓxDyΓGαx, yΓGαx, yΓx

2DyΓGαx, yΓGαx, yΓxDGαx, yΓGαx, yΓxΓy

DxΓGαx, yΓGαx, yΓy,

DxΓGαx, yΓGαx, yΓyDGαx, yΓGαx, yΓyΓx

DyΓGαx, yΓGαx, yΓxDGαx, yΓGαx, yΓxΓy.

3.47

SinceGαx, yΓyΓx Gαx, yΓyΓx xΓGαx, yΓy xΓGαx, yΓy Gαx, yΓxΓy, we obtain

DxΓGαx, yΓyDyΓGαx, yΓGαx, yΓx. 3.48

On the other hand, we also have

4DxΓyΓGαx, yΓGαx, y4DxΓGαx, yΓyΓGαx, y,

2DxΓyΓGαx, yΓGαx, yGαx, yΓGαx, yΓxΓy

2DxΓGαx, yΓyΓGαx, yyΓGαx, yΓxΓGαx, y,

2DxΓyΓGαx, yΓGαx, y2DGαx, yΓGαx, yΓxΓy

2DGαx, yΓxΓGαx, yΓy2DGαx, yΓyΓGαx, yΓx,

2DxΓyΓGαx, yΓGαx, y2DGαx, yΓGαx, yΓxΓy

DxΓGαx, yGαx, yΓxΓGαx, yΓyDyΓGαx, yGαx, yΓyΓGαx, yΓx,

2DxΓyΓGαx, yΓGαx, y2DGαx, yΓGαx, yΓxΓy

DxΓGαx, yΓGαx, yΓyDGαx, yΓGαx, yΓxΓy

DyΓGαx, yΓGαx, yΓxDGαx, yΓGαx, yΓxΓy,

2DxΓyΓGαx, yΓGαx, y

DxΓyΓGαx, yΓGαx, yDyΓxΓGαx, yΓGαx, y.

(14)

Finally using3.48we arrive atDxΓyΓGαx, yΓGαx, y DxΓyΓGαx, yΓGαx, y, but this means thatGαx, yΓGαx, yΓGαx, y 0. Hence,

Gαx, yΓGαx, yΓMΓGαx, yΓGαx, yGαx, yΓGαx, yΓGαx, yΓGαx, yΓM0,

Gαx, yΓMΓGαx, yGαx, yΓGαx, yΓM0,

3.50

which impliesGαx, y 0. The proof is complete.

Acknowledgments

The paper was supported by Grant 01-12-10-978FR MOHE Malaysia. The authors are thankful to the referee for valuable comments.

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