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,}

_{Akhil Chandra Paul,}

**1**

**and Isamiddin S. Rakhimov**

**2**

*1 _{Department of Mathematics, University of Rajshahi, Rajshahi 6205, Bangladesh}*

*2 _{Department 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 semiprime*Γ-ring satisfying the condition*aαbβcaβbαc*for all*a, b, c*∈

*M, α, β*∈Γ, and let*D*:*M* → *M*be an additive mapping such that*Dxαx* *Dxαxxαdx*for

all*x*∈*M, α*∈Γ*and 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 mapping*G* :*R* → *R*of a ring*R*with
the property*Gx*2 _{GxxxDx}_{for some derivation}_{D}_{of}_{R}_{. They prove that}_{G}_{is a Jordan}

generalized derivation.

Aydin3studied generalized derivations of prime rings. The author proved that if*I*is
an ideal of a noncommutative prime ring*R*,*a*is a fixed element of*R*and*F*is an generalized
derivation on*R*associated with a derivation*d*then the condition*Fx, a *0 orFx, a 0
for all*x*∈*I*implies*dx 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

*R*associated with a derivation*d*then*dxy, z * 0 for all*x, y, z*∈*R*and*d*is central. They
characterized a decomposition of*R*relative to the generalized derivations.

Atteya6proved that if *U*is nonzero ideal of a semiprime ring*R* and *R* admits a
generalized derivation*D* such that*Dxy*−*xy* ∈ *ZR*then *R*contains a nonzero central
ideal.

Rehman7,8studied the commutativity of a ring*R*by means of generalized
deriva-tions acting as homomorphisms and antihomomorphisms.

In this paper, we prove the following results

Let*M*be a 2-torsion-free semiprimeΓ-ring satisfying the following assumption:

*aαbβcaβbαc* ∀a, b, c∈*M, α, β*∈Γ, ∗

and *D* : *M* → *M* be an additive mapping. If there exists a derivation*d*of *M* such that

*Dxαx Dxαxxαdx*for all*x*∈*M*,*α*∈Γ, then*D*is a Jordan generalized derivation.

**2. Preliminaries**

Let*M*andΓbe additive abelian groups.*M*is called aΓ-ring if there exists a mapping*M*×
*M*×*M* → *M*such that for all*a, b, c*∈*M*,*α, β*∈Γthe following conditions are satisfied:

i*aβb*∈*M*,

ii abαc*aαcbαc*,*aαβbaαbaβb*,*aαbc aαbaαc*,aαbβc*aαbβc*.

For any*a, b*∈*M*and for*α, β*∈Γthe expressions*aαb−bαa*is denoted bya, b* _{α}*and

*αaβ*−βaα

are denoted byα, β* _{a}*. Then one has the following identities:

*aβb, c _{α}aβb, cα* a, c

*αβba*

*β, α _{c}b,*

*a, bβc _{α}bβa, cα* a, b

*αβcb*

*β, α _{a}c,* 2.1

for all*a, b, c*∈*M*and for all*α, β*∈Γ. Using the assumption∗the above identities reduce to

*aβb, c _{α}aβb, c_{α}* a, c

_{α}βb,

*a, bβc _{α}bβa, cα* a, b

*αβc,*

2.2

for all*a, b, c*∈*M*and*α*,*β*∈Γ.

Further*M*stands for a primeΓ-ring with center*ZM*. The ring*M*is*n*-torsion-free
if *nx* 0,*x* ∈ *M*implies *x* 0, where*n*is a positive integer,*M*is prime if *aΓMΓb* 0
implies*a*0 or*b*0, and it is semiprime if*aΓMΓa*0 implies*a*0. An additive mapping

*T* : *M* → *M*is called a leftrightcentralizer if*T*xαy *T*xαyTxαy *xαTy*for

*x, y*∈*M*,*α*∈Γand it is called a Jordan leftrightcentralizer if

*T*xαx *T*xαxTxαx *xαT*x ∀x∈*M, α*∈Γ. 2.3

*x, y* ∈ *M*,*α* ∈ Γand it is called a Jordan derivation if *Dxαx * *DxαxxαDx*for all

*x* ∈ *M*,*α* ∈ Γ. A derivation*D* is inner if there exists*a* ∈ *M*, such that*Dx * *aαx*−*xαa*

holds for all*x*∈*M*,*α*∈Γ. Every derivation is a Jordan derivation. The converse is in general
not true. An additive mapping*D* : *M* → *M*is said to be a generalized derivation if there
exists a derivation*d* : *M* → *M*such that *Dxαy * *Dxαyxαdy*for all*x, y* ∈ *M*,

*α* ∈ Γ. The maps of the form*x* → *aαxxαb*where*a*,*b*are fixed elements in*M*and for
all *α* ∈ Γ called the generalized inner derivation. An additive mapping*D* : *M* → *M* is
said to be a Jordan generalized derivation if there exists a derivation *d* : *M* → *M* such

that*Dxαx * *Dxαxxαdx*for all*x* ∈ *M*,*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 semiprime*Γ-ring. If

*a, b*∈

*Mare such thataαxβb*

*0 for allx*∈

*M,*

*α, β*∈Γ, then*aαbbαa0.*

*Proof. Letx*∈*M*. Then

*aαbβxγaαbaαbβxγaαb*0*,*

*bαaβxγbαabαaβxγbαa*0*.* 3.1

By semiprimeness of*M*with respect to*β, γ* ∈Γ, it follows that*aαbbαa*0.

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

*θx, yαwβϕx, y 0 for allx, y, w*∈*M, thenθx, yαwβϕu, v 0 for allx, y, u, v, w*∈*M,*

*α, β*∈Γ.

*Proof. First we replacex*with*xu*in the relation*θx, yαwβϕx, y *0, and use the

biaddi-tivity of the*θ*and*ϕ*. Then we have

*θxu, yαwβϕxu, y*0⇒*θx, yαwβϕu, y*−θ*u, yαwβϕx, y.* 3.2

Then

*θx, yαwβϕu, yγzδθx, yαwβϕu, y*

−*θu, yαwβϕx, yγzδθu, yαwβϕu, y*0*.* 3.3

Hence*θx, yαwβϕu, y *0 by semiprimeness of*M*with respect to*γ, δ*∈Γ.

* Lemma 3.3. LetMbe a semiprime*Γ-ring satisfying the assumption∗

*andabe a fixed element of*

*M. Ifaβx, y _{α}0 for allx, y*∈

*M,α, β*∈Γ, then

*a*∈

*ZM.*

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

z, a*αβxδz, aαzαaβxδz, aα*−*aαzβxδz, aα*

*zαaβz, xαaδ*−*zαaβz, xδαa*−*aαz, zδxαaβaαz, zδxβαa.*

3.4

Since *aβx, y _{α}* 0 for all

*x, y*∈

*M*,

*α, β*∈ Γ, we get z, a

*0. By the semiprimeness of*

_{α}βxδz, a_{α}*M*we getz, a

*0 for all*

_{α}*α*∈Γ. Hence

*a*∈

*ZM*.

* Lemma 3.4. LetMbe a*Γ-ring satisfying the condition∗

*andD*:

*M*→

*Mbe a Jordan generalized*

*derivation with the associated derivationd. Letx, y, z*∈*Mandα, β*∈Γ. Then

i*Dxαyyαx DxαyDyαxxαdy yαdx.*

*In particular, ifMis 2-torsion-free, then*

ii*Dxαyβx Dxαyβxxβdyβx,*

iii*Dxαyβzzβyαx DxαyβzDzβyαxxαdyβz zβdyαx.*

*Proof.* iWe have*Dxαx Dxαxxαdx*, for all*x*∈*M*,*α*∈Γ. Then replacing*x*by*xy*,

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

iiReplace*y*by*xβyyβx*in 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.

Using the assumption∗, we conclude that

*Dxαxβyyβxαx*2*DxαyβxDxαxβyDxαyβxDxβyαxDyβxαx*

*xβdyαxyβdxαxxαdxβyyβx*

*xβyyβxαdx,* ∀x, y, z∈*M, α, β*∈Γ.

3.7

Again, replacing*x*by*xβx*in3.5

*DxβxαyyαxβxDxβxαyDyαxβxxβxαdy*

*yαdxβx,* ∀x, y∈*M, α, β*∈Γ,

⇒*DxβxαyyαxβxDxβxαyxβdxαyDyαxβxxβxαdy*

*yαdxβxyαxβdx,* ∀x, y∈*M, α, β*∈Γ.

3.8

Adding both sides 2*Dxαyβx*, we get,

⇒*Dxβxαyyαxβx*2*Dxαyβx*

*DxβxαyxβdxαyDyαxβxxβxαdy*

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

3.9

Comparing3.7and3.9we obtain,

2*Dxαyβx*2*Dxαyβxxαyβdx xαdyβx.* 3.10

Since*M*is 2-torsion-free, it gives

*DxαyβxDxαyβxxαdyβx.* 3.11

iiiReplace*x*for*xz*in3.12, we get,

*D*x*zαyβxz*

*Dxzαyβxz xzαdyβxz*

⇒*Dxαyβxxαyβzzαyβxzαyβz*

*DxαyβxDzαyβxDxαyβzDzαyβzxαdyβx*

*xαdyβzzαdyβxyβz*

⇒*DxαyβxzαyβzDxαyβzzαyβx*

*DxαyβxDzαyβzxαdyβxzαdyβz*

⇒*Dxαyβzzαyβx*

*DxαyβzDzαyβxxαdyβzzαdyβx*

⇒*Dxαyβzzβyαx*

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

3.12

*Definition 3.5. LetM*be aΓ-ring and*D* :*M* → *M*be a Jordan generalized derivation with

the associated derivation*d*. Define*Gαx, y * *Dxαy*−*Dxαy*−*xαdy*for all*x, y* ∈ *M*

and*α*∈Γ.

**Lemma 3.6. The function**Gαx, yhas the following properties:

i*Gαx, y Gαy, x 0.*

ii*Gαx, yz Gαx, y Gαx, z.*

iii*Gαxy, z Gαx, z Gαy, z.*

iv*Gαβx, y Gαx, y Gβx, y.*

*Proof. The results easily follow from*Lemma 3.4i.

*Remark 3.7.* *D*is a generalized derivation if and only if*Gαx, y *0 for all*x, y*∈*M*,*α*∈Γ.

* Theorem 3.8. LetMbe a 2-torsion-free semiprime*Γ-ring satisfying the condition∗. Let

*x, y, z*∈

*Mandβ, δ*∈Γ. Then

i*Gαx, yβzδx, y _{α}0 for allz*∈

*M,*

ii*Gαx, yβx, y _{α}0.*

*Proof.* iFromLemma 3.4iiiwe get

*DxαyβzzβyαxDxαyβzDzβyαxxαdyβzzβdyαx.* 3.13

We set

*Axαyβzδyαxyαxβzδxαy.* 3.14

Since

*DA Dxαyβzδyαxyαxβzδxαy*

*Dxαyβzδyαxyαxβzδxαy*

*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,

*Dxαy*−*Dxαy*−*xαdyβzδyαxDyαx*−*Dyαx*−*yαdxβzδxαy*0*.*

⇒*Gαx, yβzδyαxGαy, xβzδxαy*0*,*

⇒*Gαx, yβzδyαx*−*Gαx, yβzδxαy*0*,*

⇒*Gαx, yβzδy, x _{α}*0

*,*

⇒*Gαx, yβzδx, y _{α}*0

*,*for

*x, y*∈

*M, α, β, δ*∈Γ.

3.17

iiAccording toLemma 3.4iiwe have,

*DxαyβzDxαyβzxαdyβz.* 3.18

Replace*z*by*xαy*in3.18we find

*DxαyβxαyDxαyβxαyxαdyβxαy*

⇒*DxαyβxαyDxαyβxαyxαdyβxαy*

⇒*Dxαyβxαyxαyβdxαy*−*Dxαyβxαy*−*xαdyβxαy*0

⇒ *Dxαy*−*Dxαy*−*xαdyβxαyxαyβdxαy*−*xαyβdxαy*0

⇒*Gαx, yβxαy*0*.*

3.19

Similarly*Gαx, yβyαx*0. Therefore,

*Gαx, yβxαy*−*Gαx, yβyαxGαx, yβx, y _{α}*0

*.*3.20

**Lemma 3.9.** *Gαx, y*∈*ZM, for allx, y*∈*M,α*∈Γ.

*Proof. From*Theorem 3.8ii, we have *Gαx, yβx, y _{α}* 0. Therefore due toLemma 3.4ii

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

*D* : *M* → *Mbe a Jordan generalized derivation with associated derivationdonM. Then* *Dis a*

*generalized derivation.*

*Proof. In particular,rγGαx, y*,*Gαx, yγr* ∈*ZM*for all*r* ∈*M*,*α, γ* ∈Γ. This gives

*xαGαx, yδGαx, yβyGαx, yδGαx, yβyαx*

*yβGαx, yδGαx, yαx*

*yαGαx, yδGαx, yβx.*

3.21

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

4*DxαGαx, yδGαx, yβy*

2*DxαGαx, yδGαx, yβyGαx, yδGαx, yβyαx*

2*DxαGαx, yδGαx, yβy*2*xαdGαx, yδGαx, yβy*

2*DGαx, yδGαx, yβyαx*2*Gαx, yδGαx, yβyαdx*

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

2*xαdGαx, yδGαx, yβy*2*Gαx, yδGαx, yβyαdx*

2*DxαGαx, yδGαx, yβyDGαx, yδGαx, yβyαx*

*Gαx, yδdGαx, yβyαxDyβGαx, yδGαx, yαx*

*Gαx, yδGαx, yβdyαxyβdGαx, yδGαx, yαx*

2*xαdGαx, yδGαx, yβy*2*Gαx, yδGαx, yβyαdx.*

3.22

So we get

4*DxβGαx, yδGαx, yαy*

2*DxβGαx, yδGαx, yαyDGαx, yδGαx, yβyαx*

*Gαx, yδdGαx, yβyxDyβGαx, yδGαx, yαx*

*Gαx, yδGαx, yβdyαxyβdGαx, yδGαx, yαx*

2*xαdGαx, yδGαx, yβy*2*Gαx, yδGαx, yβyαdx,* *x, y*∈*M, α, β, δ*∈Γ.

3.23

Moreover,

4*DyβGαx, yδGαx, yαx*

2*DyβGαx, yδGαx, yαxGαx, yδGαx, yβxαy*

2*DyβGαx, yδGαx, yαx*2*yβdGαx, yδGαx, yαx*

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

2*yβdGαx, yδGαx, yαx*2*Gαx, yδGαx, yβxαdy*

2*DyβGαx, yδGαx, yαxDGαx, yδGαx, yβxαy*

*Gαx, yδdGαx, yβxαyDxβGαx, yδGαx, yαy*

*Gαx, yδGαx, yαdxβyxαdGαx, yδGαx, yβy*

2*yβdGαx, yδGαx, yαx*2*Gαx, yδGαx, yαxβdy.*

3.24

So we get

4*DyβGαx, yδGαx, yαx*

2*DyβGαx, yδGαx, yαxDGαx, yδGαx, yαxβy*

*Gαx, yδdGαx, yαxβyDxβGαx, yδGαx, yαy*

*Gαx, yδGαx, yαdxβyxαdGαx, yδGαx, yβy*

2*yβdGαx, yδGαx, yαx*2*Gαx, yδGαx, yαxβdy,*

*x, y*∈*M, 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*

which gives

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

where*ϕαx, y*stands for*Dxαyxαdy.*
On the other hand, we have

4*DxαyβGαx, yδGαx, y*4*DxαGαx, yδyβGαx, y.* 3.29

Now we use3.15and the properties of*Gαx, y*, to derive

4*DxαyβGαx, yδGαx, y*

2*DxαyβGαx, yδGαx, yGαx, yδGαx, yαxβy*

2*DxαyβGαx, yδGαx, y*2*xαyβdGαx, yδGαx, y*

2*DGαx, yδGαx, yαxβy*2*Gαx, yδGαx, yβdxαy,*

3.30

which gives

4*DxαyβGαx, yδGαx, y*2*DxαyβGαx, yδGαx, y*2*xαyβdGαx, yδGαx, y*

2*DGαx, yδGαx, yαxβy*

2*Gαx, yδGαx, yβdxαy,* *x, y*∈*M, α, β, δ*∈Γ.

3.31

Moreover,

4*DxαGαx, yδyβGαx, y*

2*DxαGαx, yδyβGαx, yyβGαx, yδxαGαx, y*

2*DGαx, yαxδGαx, yβy*2*Gαx, yαxδdGαx, yβy*

2*DGαx, yβyδGαx, yαx*2*Gαx, yβyδdGαx, yαx*

*DxαGαx, yGαx, yαxδGαx, yβy*2*Gαx, yαxδdGαx, yβy*

*DyβGαx, yGαx, yβyδGαx, yαx*2*Gαx, yβyδdGαx, yαx*

*DxαGαx, yδGαx, yβyDGαx, yδGαx, yαxβy*

*xαdGαx, yδGαx, yβyGαx, yαdxδGαx, yβy*

2*Gαx, yαxδdGαx, yβyDyβGαx, yδGαx, yαx*

*DGαx, yδGαx, yβyαxyβdGαx, yδGαx, yαx*

*Gαx, yβdyδGαx, yαx*2*Gαx, yβyδdGαx, yαx.*

So we obtain

4*DxαGαx, yδyβGαx, y*

*DxαGαx, yδGαx, yβyDGαx, yδGαx, yβxαy*

*xαdGαx, yδGαx, yβyGαx, yαdxδGαx, yβy*

2*Gαx, yαxδdGαx, yβyDyβGαx, yδGαx, yαx*

*DGαx, yδGαx, yβyαxyβdGαx, yδGαx, yαx*

*Gαx, yβdyδGαx, yαx*2*Gαx, yβyδdGαx, yαx,*

*x, y*∈*M, α, β, δ*∈Γ.

3.33

Comparing3.31and3.33, we derive

2*DxαyβGαx, yδGαx, y*

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

3.34

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

*Gαx, y * *Dxαy*−*ϕαx, y*. By the semiprimeness of*M*, we have*ϕαx, yβGαx, y * 0.

Again by the primeness of*M*, we get*Gαx, y*. The proof is complete.

It is clear that if we let the derivation*d*to 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* : *M* → *M* *be an additive*

*mapping which satisfiesDxαx Dxαxfor allx*∈*M,α*∈Γ. Then*Dis a left centralizer*

*Proof. We have*

*Dxαx Dxαx.* 3.35

If we replace*x*by*xy*, we get

*DxαyyαxDxαyDyαx.* 3.36

By replacing*y*with*xαyyαx*and using3.5, we arrive at

*DxαxαyyαxxαyyαxαxDxαxαyDxαyαxDxαyαxDyαxαx.*

3.37

But on the other hand,

Comparing3.37and3.38we obtain

*DxαyαxDxαyαx.* 3.39

If we linearize3.39in*x*, we get

*DxαyαzzαyαxDxαyαzDzαyαx.* 3.40

Now we shall compute*Dxαyαzαyαxyαxαzαxαy*in 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 mapping*Gαx, y * *Dxαy*−

*Dxαy*we arrive at

*Gαx, yαzαyαxGαy, xαzαxαy*0*.* 3.43

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

3.43we obtain

*Gαx, yαzαx, y _{α}*0

*.*3.44

Using firstLemma 3.2and thenLemma 3.1we have

*Gαx, yαzαu, vα*0*.* 3.45

Now fix some*x, y*∈*M*and usingLemma 3.3we get*Gαx, y*∈*Z*.
In particular,*rβGαx, y*,*Gαx, yβr*∈*Z*for all*r*∈*M*. This gives

*xβGαx, yδGαx, yβyGαx, y*ΓGα*x, y*ΓyΓx

*yΓGαx, y*ΓGα*x, y*Γx

*yΓGαx, y*ΓGα*x, y*Γx.

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

2*DxΓGαx, y*ΓGα*x, y*Γy*Gαx, y*ΓGα*x, y*ΓyΓx

2*DyΓGαx, y*ΓGα*x, y*Γx*Gαx, y*ΓGα*x, y*ΓxΓy*,*

2*DxΓGαx, y*ΓGα*x, y*Γy2*DGαx, y*ΓGα*x, y*ΓyΓx

2*Dy*ΓGα*x, y*ΓGα*x, y*Γx2*DGαx, y*ΓGα*x, y*ΓxΓy,

2*DxΓGαx, y*ΓGα*x, y*Γy*DGαx, y*ΓGα*x, y*Γy*yΓGαx, y*ΓGα*x, y*Γx

2*Dy*ΓGα*x, y*ΓGα*x, y*Γx*DGαx, y*ΓGα*x, y*Γx*xΓGαx, y*ΓGα*x, y*Γy,

2*DxΓGαx, y*ΓGα*x, y*Γy*DGαx, y*ΓGα*x, y*ΓyΓx*Dy*ΓGα*x, y*ΓGα*x, y*Γx

2*Dy*ΓGα*x, y*ΓGα*x, y*Γx*DGαx, y*ΓGα*x, y*ΓxΓy

*DxΓGαx, y*ΓGα*x, y*Γy,

*DxΓGαx, y*ΓGα*x, y*Γy*DGαx, y*ΓGα*x, y*ΓyΓx

*Dy*ΓGα*x, y*ΓGα*x, y*Γx*DGαx, y*ΓGα*x, y*ΓxΓy.

3.47

Since*Gαx, yΓyΓx* *Gαx, yΓyΓx* *xΓGαx, yΓy* *xΓGαx, yΓy* *Gαx, yΓxΓy*, we
obtain

*DxΓGαx, y*Γy*Dy*ΓGα*x, y*ΓGα*x, y*Γx. 3.48

On the other hand, we also have

4*DxΓyΓGαx, y*ΓGα*x, y*4*DxΓGαx, y*ΓyΓGα*x, y,*

2*DxΓyΓGαx, y*ΓGα*x, yGαx, y*ΓGα*x, y*ΓxΓy

2*DxΓGαx, y*ΓyΓGα*x, yyΓGαx, y*ΓxΓGα*x, y,*

2*DxΓy*ΓGα*x, y*ΓGα*x, y*2*DGαx, y*ΓGα*x, y*ΓxΓy

2*DGαx, y*ΓxΓGα*x, y*Γy2*DGαx, y*ΓyΓGα*x, y*Γx,

2*DxΓy*ΓGα*x, y*ΓGα*x, y*2*DGαx, y*ΓGα*x, y*ΓxΓy

*DxΓGαx, yGαx, y*ΓxΓGα*x, y*Γy*DyΓGαx, yGαx, y*ΓyΓGα*x, y*Γx,

2*DxΓy*ΓGα*x, y*ΓGα*x, y*2*DGαx, y*ΓGα*x, y*ΓxΓy

*DxΓGαx, y*ΓGα*x, y*Γy*DGαx, y*ΓGα*x, y*ΓxΓy

*Dy*ΓGα*x, y*ΓGα*x, y*Γx*DGαx, y*ΓGα*x, y*ΓxΓy,

2*DxΓy*ΓGα*x, y*ΓGα*x, y*

*DxΓyΓGαx, y*ΓGα*x, yDy*ΓxΓGα*x, y*ΓGα*x, y.*

Finally using3.48we arrive at*DxΓyΓGαx, yΓGαx, y DxΓyΓGαx, yΓGαx, y*, but
this means that*Gαx, yΓGαx, yΓGαx, y *0. Hence,

*Gαx, y*ΓGα*x, y*ΓMΓGα*x, y*ΓGα*x, yGαx, y*ΓGα*x, y*ΓGα*x, y*ΓGα*x, y*ΓM0*,*

*Gαx, y*ΓMΓGα*x, yGαx, y*ΓGα*x, y*ΓM0*,*

3.50

which implies*Gαx, 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.

**References**

1 *B. Hvala, “Generalized derivations in rings,” Communications in Algebra, vol. 26, no. 4, pp. 1147–1166,*

1998.

2 M. N. Daif and M. S. Tammam El-Sayiad, “On generalized derivations in semiprime rings,”

*International Journal of Contemporary Mathematical Sciences, vol. 2, no. 29–32, pp. 1481–1486, 2007.*

3 *N. Aydin, “A note on generalized derivations of prime rings,” International Journal of Algebra, vol. 5,*

no. 1–4, pp. 17–23, 2011.

4 Y. C¸ even and M. A. ¨*Ozt ¨urk, “On Jordan generalized derivations in gamma rings,” Hacettepe Journal of*

*Mathematics and Statistics, vol. 33, pp. 11–14, 2004.*

5 *F. Ali and M. A. Chaudhry, “On generalized derivations of semiprime rings,” International Journal of*

*Algebra, vol. 4, no. 13–16, pp. 677–684, 2010.*

6 *M. J. Atteya, “On generalized derivations of semiprime rings,” International Journal of Algebra, vol. 4,*

no. 9–12, pp. 591–598, 2010.

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*Okayama University, vol. 44, pp. 43–49, 2002.*

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