Volume 2012, Article ID 625968,14pages doi:10.1155/2012/625968

*Research Article*

**On Prime-Gamma-Near-Rings with**

**Generalized Derivations**

**Kalyan Kumar Dey,**

**1**

_{Akhil Chandra Paul,}

_{Akhil Chandra Paul,}

**1**

**and Isamiddin S. Rakhimov**

**2**

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

*2 _{Department of Mathematics, Faculty of Science and Institute for Mathematical Research (INSPEM),}*

*Universiti Putra Malaysia, 43400 Serdang, Malaysia*

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

Received 26 January 2012; Accepted 19 March 2012

Academic Editor: B. N. Mandal

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*N* be a 2-torsion free prime Γ-near-ring with center *ZN*. Let *f, d* and *g, h* be two
generalized derivations on*N*. We prove the following results:iif*fx, yα* 0 or*fx, yα*
±*x, y _{α}* or

*f*2

_{}

_{x}_{}

_{∈}

_{Z}_{}

_{N}_{}

_{for all}

_{x, y}_{∈}

_{N}_{,}

_{α}_{∈}

_{Γ}

_{, then}

_{N}_{is a commutative}

_{Γ}

_{-ring.}

_{}

_{ii}

_{}

_{If}

_{a}_{∈}

_{N}and*fx, a _{α}* 0 for all

*x*∈

*N*,

*α*∈Γ, then

*da*∈

*ZN*.iiiIf

*fg, dh*acts as a generalized derivation on

*N*, then

*f*0 or

*g*0.

**1. Introduction**

The derivations inΓ-near-rings have been introduced by Bell and Mason1. They studied

basic properties of derivations in Γ-near-rings. Then As¸ci 2 obtained commutativity

conditions for a Γ-near-ring with derivations. Some characterizations of Γ-near-rings and

regularity conditions were obtained by Cho3. Kazaz and Alkan4introduced the notion

of two-sidedΓ-α-derivation of aΓ-near-ring and investigated the commutativity of a prime

and semiprimeΓ-near-rings. Uc¸kun et al.5worked on primeΓ-near-rings with derivations

and they found conditions for a Γ-near-ring to be commutative. In6 Dey et al. studied

commutativity of primeΓ-near-ring with generalized derivations.

In this paper, we obtain the conditions of a primeΓ-near-ring to be a commutative

Γ-ring. If*a*∈*N, and*fx, a* _{α}*0 for all

*x*∈

*N, α*∈Γ, then

*d*is central. Also we prove that if

**2. Preliminaries**

AΓ-near-ring is a tripleN,,Γ, where

i N,is a groupnot necessarily abelian;

ii Γis a nonempty set of binary operations on*N*such that for each*α*∈Γ,N,, αis

a left near-ring;

iii*xαyβz xαyβz, for allx, y, z*∈*N*and*α, β*∈Γ.

We will use the wordΓ-near-ring to mean leftΓ-near-ring. For a near-ring*N, the set*

*N0*{x∈*N*: 0αx0, α∈Γ}is called the zero-symmetric part of*N. A*Γ-near-ring*N*is said

to be zero-symmetric if*NN*0. Throughout this paper,*N*will denote a zero symmetric left

Γ-near-ring with multiplicative centre*ZN. Recall that a*Γ-near-ring*N*is prime if*xΓNΓy*

0 implies *x* 0 or *y* 0. An additive mapping *d* : *N* → *N* is said to be a derivation

on *N* if *dxαy * *xαdy dxαy* for all *x, y* ∈ *N, α* ∈ Γ, or equivalently, as noted in

1, that*dxαy * *dxαyxαdy*for all*x, y* ∈ *N, α* ∈ Γ. Further, an element*x* ∈ *N* for

which*dx * 0 is called a constant. For*x, y* ∈ *N, α* ∈ Γ, the symbolx, y* _{α}* will denote the

commutator*xαy*−*yαx, while the symbol*x, ywill denote the additive-group commutator

*xy*−*x*−*y. An additive mappingf* :*N* → *N*is called a generalized derivation if there exits

a derivation*d*of*N*such that*fxαy f*xαy*xαdy*for all*x, y*∈*N, α*∈Γ. The concept

of generalized derivation covers also the concept of a derivation.

**3. Derivations on**

### Γ

**-Near-Rings**

In this section we prove that a few subsidiary resultsLemmas3.1,3.2,3.4,3.8,3.9,3.10and

3.11to use them for proving of our main resultsTheorems3.3,3.5,3.6,3.12and3.13.

**Lemma 3.1. Let***dbe an arbitrary derivation on a*Γ*-near-ringN. ThenN* *satisfies the following*

*partial distributive law:*aαdb *daαbβcaαdbβcdaαbβcand*daαb*aαdbβc*

*daαbβcaαdbβcfor alla, b, c*∈*N, α, β*∈Γ*.*

*Proof. For all* *a, b, c* ∈ *N, α, β* ∈ Γ, we get*daαbβc * *aαbβdc aαdb daαbβc*

and *daαbβc * *aαdbβc daαbβc* *aαbβdc dbβc daαbβc* *aαbβdc *

*aαdbβcdaαbβc. Equating these two relations for* *daαbβc* now yields the required

partial distributive law.

* Lemma 3.2. Letdbe a derivation on a*Γ

*-near-ringNand supposeu*∈

*Nis not a left zero divisor. If*

u, du*α*0, α∈Γ*, then*x, u*is a constant for everyx*∈*N.*

*Proof. Fromuαux uαuuαx, for allx*∈*N,α*∈Γ, we obtain*uαdux duαux *

*uαdu duαuuαdx duαx, which reducesuαdx duαuduαuuαdx, for*

all*α*∈Γ.

Since*duαuuαdu,α*∈Γ, this equation is expressible as*uαdx du*−*dx*−

*du *0*uαdx, u. Thusdx, u *0.

* Theorem 3.3. LetNbe a*Γ

*-near-ring having no nonzero divisors of zero. IfN*

*admits a nontrivial*

*Proof. Letc*be any additive commutator. Then*c*is a constant byLemma 3.2. Moreover, for

any*w*∈*N,α*∈Γ,*wαc*is an additive commutator, hence also a constant. Thus, 0*dwαc *

*wαdc dwαc*and*dwαc* 0, for all*α*∈ Γ. Since*dw/*0 for all*w* ∈ *N, we conclude*

that*c*0.

* Lemma 3.4. LetNbe a prime*Γ

*-near-ring.*

i*Ifz*∈*ZN*− {0}*, thenzis not a zero divisor inN.*

ii*IfZN*− {0}*contains an elementzfor whichzz*∈*ZN, then*N,*is abelian.*

iii*Letdbe a nonzero derivation onN. ThenxΓdN *{0}*impliesx0, anddNΓx*{0}

*impliesx0.*

iv*IfNis 2-torsion free anddis a derivation onNsuch thatd*2 _{}_{0, then}_{d}_{}_{0.}

*Proof.* iIf*z*∈*ZN*− {0}and*zαx*0,*x*∈*N,α*∈Γ, then*zαrβx*0,*x,r* ∈*N,α*∈Γ. Thus

we get*zΓNΓx*0, by primeness of*N,x*0.

iiLet*z*∈*ZN*− {0}be an element such that*zz*∈*ZN, and letx, y*∈*N, α*∈Γ.

Since*zz*is distributive, we getx *yαz* *z xαz* *z * *yαz* *z xαz* *xαz*

*yαz* *yαz* *zαx* *x* *y* *y.*

On the other hand,x*yαzz xyαz* x*yαzzαxyxy. Thus,*

*xxyyxyxy*and therefore*xyyx. Hence*N,is abelian.

iii Let *xΓdN * 0, and let *r, s* be arbitrary elements of *N* and *α, β* ∈ Γ. Then

0 *xαdrβs * *xαrβds xαdrβs* *xαrβds. Thus* *xΓNΓdN * {0}, and since

*dN/*{0}, x0.

A similar argument works if *dNΓx* {0}, since Lemma 3.1 provides enough

distributivity to carry it through.

ivFor arbitrary *x, y* ∈ *N, α* ∈ Γ, we have 0 *d*2_{xαy } _{dxαdy }_{dxαy }

*xαd*2_{y } _{dxαdy }_{dxαdy }* _{d}*2

_{xαy}

_{}

_{2dxαdy. Since}

_{N}_{is 2-torsion free,}

*dxαdy * 0, x, y ∈ *N, α* ∈ Γ. Thus *dxΓdN * {0} for each *x* ∈ *N, and* iiiyields

*da *0. Thus*d*0.

* Theorem 3.5. If a prime*Γ

*-near-ringNadmits a nontrivial derivationdfor whichdN*∈

*ZN,*

*then*N,*is abelian. Moreover, ifNis 2-torsion free, thenNis a commutative*Γ*-ring.*

*Proof. Letc* be an arbitrary constant, and let*x* be anon-constant. Then*dxαc * *xαdc *

*dxαc* *dxαc* ∈ *ZN,α* ∈Γ. Since*dx* ∈ *ZN*− {0}, it follows easily that*c* ∈ *ZN.*

Since*cc*is a constant for all constants*c, it follows from*Lemma 3.4iithatN,is abelian,

provided that there exists a nonzero constant.

Assume, then, that 0 is the only constant. Since*d*is obviously commuting, it follows

fromLemma 3.2that all*u*which are not zero divisors belong to the center ofN,, denoted

by*ZN. In particular, ifx /*0,*dx*∈*ZN. But then for ally*∈*N,dydx−dy−dx *

*dy, x *0, hencey, x 0.

Now we assume that *N* is 2-torsion free. By Lemma 3.1, aαdb *daαbβc*

*aαdbβcdaαbβc*for all*a, b, c* ∈ *N, α, β* ∈ Γ, and using the fact that *daαb* ∈ *ZN,*

*α*∈ Γ, we get*cαaβdb cαdaβb* *aαdbβcaαdbβc, α, β* ∈ Γ. SinceN,is abelian

and*dN*⊆*ZN, this equation can be rearranged to yielddbαc, a _{β}daαb, c_{β}*for all

*a, b, c*∈*N, α, β*∈Γ.

Suppose now that*N* is not commutative. Choosing*b, c* ∈ *N, with*b, c* _{β}/*0,

*β*∈ Γ,

and letting*a* *dx, we getd*2_{xαb, c}

elements*d*2_{x}_{cannot be a nonzero divisor of zero, we conclude that}* _{d}*2

_{x }

_{0 for all}

_{x}_{∈}

_{N.}But byLemma 3.4iv, this cannot happen for nontrivial*d.*

**Theorem 3.6. Let***N* *be a prime* Γ*-near-ring admitting a nontrivial derivation* *d* *such that*

dx, dy_{α}*0 for all* *x, y* ∈ *N, α* ∈ Γ*. Then* N, *is abelian. Moreover, ifN* *is 2-torsion*

*free, thenNis a commutative*Γ*-ring.*

*Proof. By*Lemma 3.4ii, if both*z*and*zz*commute element-wise with*dN, thenzαdc *

0, α ∈Γ, for all additive commutators*c. Thus, takingz* *dx, we getdxαdc * 0 for all

*x* ∈*N, α* ∈Γ, so*dc * 0 byLemma 3.4iii. Since*wαc*is also an additive commutator for

any*w* ∈ *N,α*∈Γ, we have*dwαc *0 *dwαc, and another application of*Lemma 3.4iii

gives*c*0.

Now we assume that *N* is 2-torsion free. By the partial distributive law,

*ddxαyβdz * *dxαdyβdz d*2_{xαyβdz} _{for all} _{x, y, z}_{∈} _{N, α, β}_{∈} _{Γ, hence,}

*d*2_{xαyβdz } _{ddxαyβdz}_{−}_{dxαdyβdz }_{dxαdyβdz }_{dzαddxβy}_{−}

*dxβdy dzαd*2_{xβy}_{}* _{d}*2

_{xαdzβy, α, β}

_{∈}

_{Γ. Thus}

*2*

_{d}_{xαyβdz}

_{−}

_{dzβy }_{0 for}

all*x, y z*∈*N,α, β*∈Γ.

Replacing*yδt, δ*∈Γ, we obtain*d*2_{xαyδtβdz }* _{d}*2

_{xαdzβyδt}

_{}

*2*

_{d}_{xαyβdzδt,}

for all*x, y, z, t*∈*N,α, β, δ*∈Γ, so that*d*2xαyβt, dz* _{δ}*0 for all

*x, y, z, t*∈

*N,α, β, δ*∈Γ.

The primeness of*N*shows that either*d*2 _{} _{0 or}_{dN}_{⊆} _{ZN, and since the first of these}

conditions is impossible byLemma 3.4iv, the second must hold and *N*is a commutative

Γ-ring byTheorem 3.5.

*Definition 3.7. LetN*be aΓ-near-ring and*d*a derivation of*N. An additive mappingf*:*N* →

*N*is said to be a right generalized derivation of*N*associated with*d*if

*fxαyfxαyxαdy* ∀x, y∈*R, α*∈Γ, 3.1

and*f*is said to be a left generalized derivation of*N*associated with*d*if

*fxαydxαyxαfy* ∀x, y∈*R, α*∈Γ. 3.2

Finally,*f* is said to be a generalized derivation of*N*associated with*d*if it is both a left and

right generalized derivation of*N*associated with*d.*

* Lemma 3.8. Letfbe a right generalized derivation of a*Γ

*-near ringNassociated withd. Then*

i*fxαy xαdy f*xαy*for allx, y*∈*N, α*∈Γ*;*

ii*fxαy xαf*y *dxαyfor allx, y*∈*N, α*∈Γ*.*

*Proof.* iFor any*x, y*∈*N, α*∈Γ, we get

*fxαyyf*xα*yyxαdyyf*xαy*fxαyxαdyxαdy,*

*fxαyxαyfxαyxαdyfxαyxαdy.*

Comparing these two expressions, we obtain

*fxαyxαdyxαdyfxαy* ∀x, y∈*N, α*∈Γ, 3.4

and so,

*fxαyxαdyfxαy* ∀x, y∈*N, α*∈Γ. 3.5

iiIn a similar way.

* Lemma 3.9. Letfbe a right generalized derivation of a*Γ

*-near ringNassociated withd. Then*

i fxαy*xαdyβzfxαyβzxαdyβz, for allx, y*∈*N,α, β*∈Γ*.*

ii dxαy*xαf*yβz*dxαyβzxαfyβz, for allx, y*∈*N,α, β*∈Γ*.*

*Proof.* iFor any*x, y, z*∈*N,α, β*∈Γ, we get*fxαyβz f*xαyβz*xαyβdz.*
On the other hand,

*fxαyβzf*xαyβz*xαdyβzfxαyβzxαdyβzxαyβdz.* 3.6

From these two expressions of*fxαyβz, we obtain that, for allx, y, z*∈*N,α, β*∈Γ,

*fxαyxαdyβzfxαyβzxαdyβz.* 3.7

iiThe proof is similar.

* Lemma 3.10. LetNbe a prime*Γ

*-near-ring,fa nonzero generalized derivation ofNassociated with*

*the nonzero derivationdanda*∈*N.*i*IfaΓfN 0, thena0.*ii*IffNΓa0, thena0.*

*Proof.* iFor any*x, y*∈*N, α, β*∈Γ, we get 0*aβf*xαy *aβf*xαyaβxαdy *aβxαdy.*

Hence*aΓNΓdN *0. Since*N*is a primeΓ-near-ring and*d /*0, we obtain*a*0.

iiA similar argument works if*f*NΓa0.

* Lemma 3.11. LetNbe a prime*Γ

*-near-ring. Letfbe a generalized derivation ofNassociated with*

*the nonzero derivationd. IfNis a 2-torsion free*Γ*-near-ring andf*2_{}_{0, then}_{f}_{}_{0.}

*Proof.* iFor any*x, y*∈*N,α*∈Γ, we get

0*f*2_{xαy}_{}_{f}_{f}_{xαy}_{}_{f}_{fxαy}_{}_{xαd}_{y}_{}* _{f}*2

_{xαy}

_{}

_{2fxαd}

_{y}_{}

*2*

_{xαd}

_{y}_{.}3.8

By the hypothesis,

2fxαd*yxαd*2_{y}_{}_{0} _{∀x, y}_{∈}_{N, α}_{∈}_{Γ.} _{3.9}

ByLemma 3.9ii, we obtain that*d*2_{N } _{0 or} _{f}_{} _{0. If}* _{d}*2

_{N }

_{0 then}

_{d}_{}

_{0 from}

Lemma 3.4iv, a contradiction. So we find*f*0.

* Theorem 3.12. LetNbe a prime*Γ

*-near-ring with a nonzero generalized derivationfassociated with*

*d. IffN*⊆*ZN, then*N,*is abelian. Moreover, ifNis 2-torsion free, thenNis commutative*

Γ*-ring.*

*Proof. Suppose that* *a* ∈ *N, such thatfa/*0. So,*f*a ∈ *ZN*− {0} and *fa fa* ∈

*ZN*− {0}. For all*x, y*∈*N,α*∈Γ, we havex*yαfafa fafaαxy.*

That is,*xαf*a *xαf*a *yαfa yαfa faαxfaαxfaαyfaαy, for*

all*x, y*∈*N,α*∈Γ.

Since*fa*∈*ZN, we getfaαxfaαyfaαyfaαx, and so,faαx, y *0

for all*x, y*∈*N, α*∈Γ.

Since*fa*∈*ZN*− {0}and*N*is a primeΓ-near-ring, it follows thatx, y 0, for all

*x, y*∈*N. Thus*N,is abelian.

Using the hypothesis, for any *x, y, z* ∈ *N, α, β* ∈ Γ, *zαfxβy * *fxβyαz. By*

Lemma 3.4ii, we have*zαdxβyzαxfy * *dxαyβzxαfyβz. UsingfN*⊂ *ZN*

andN,being abelian, we obtain that

*zαdxβy*−*dxαyβz* *x, zαβf*

*y,* ∀x, y∈*N, α, β*∈Γ. 3.10

Substituting*fz*for*z*in3.10, we get*fzβdx, y _{α}*0 for all

*x, y*∈

*N, α, β*∈Γ.

Since*f*z∈*ZN*and f a nonzero generalized derivation with associated with*d, we*

get*dN*⊂*ZN. So,N*is a commutativeΓ-ring byTheorem 3.3.

* Theorem 3.13. LetNbe a prime*Γ

*-near-ring with a nonzero generalized derivationfassociated with*

*d. If*fN, fN_{α}*0,α*∈Γ*, then*N,*is abelian. Moreover, ifNis 2-torsion free, thenNis*

*commutative*Γ*-ring.*

*Proof. By the same argument as in*Theorem 3.12, it is shown that if both*z*and*zz*commute

elementwise with*fN, then we have*

*zαfx, y*0 ∀x, y∈*N, α*∈Γ. 3.11

Substituting*f*t, t ∈ *N* for*z* in 3.11, we get*ftαf*x, y 0, *α* ∈ Γ. ByLemma 3.9i,

we obtain that *fx, y * 0 for all*x, y* ∈ *N,α* ∈ Γ. For any*w* ∈ *N,* *β* ∈ Γ, we have 0

*fwβx, wβy fwβx, y dwβx, ywβfx, y*and so, we obtain*dwβx, y *0, for

any*w*∈*N,β*∈Γ. FromLemma 3.4iii, we getx, y 0 for any*x, y*∈*N.*

Now we assume that*N*is 2-torsion free. By the assumptionfN, fN* _{α}*0,

*α*∈Γ,

we have

*fzαffxβyffxβyαf*z ∀x, y, z∈*N, α, β*∈Γ. 3.12

Hence we get

*fzαdfxβyfzαf*xβf*ydfxαyβfz fxαfyβfz,*

and so,

*fzαdfxβydfxαyβfz* ∀x, y, z∈*N, α, β*∈Γ. 3.14

If we take*yδw*instead of*y*in3.14, then

*dfxαyδwβf*z *fzαdfxβyδwdfxαyδf*zβw

∀x, y, z∈*N, α, β, δ*∈Γ, 3.15

and so,

*df*x*αyδwβf*z−*dfxαyδf*zβw*dfxαyδfz, w _{δ}*0

∀x, y, z∈*N, α, β, δ*∈Γ. 3.16

Thus we get*dfxΓNΓfz, w _{δ}* 0, for all

*x, y, z*∈

*N,α, β, δ*∈ Γ. Since

*N*is a prime

Γ-near-ring, we have*dfN *0 or*fN*⊂*ZN. Let us assume thatdfN *0. Then

0*dfxαyddxαyxαfy* 3.17

and so,

*d*2xαy*dxαdydxαfy*0, ∀x, y∈*N, α*∈Γ. 3.18

Replacing*y*by*yβz,β*∈Γ, in3.18, we get

0*d*2xαyβz*dxαdyβzdxαfyβz*

*d*2xαyβz*dxαdyβzdxαyβdz dxαfyβzdxαyβdz*

*d*2xαy*dxαdydxαfyβz*2dxαyβdz ∀x, y, z∈*N, α, β*∈Γ.

3.19

Using3.18and*N*being 2-torsion freeΓ-near-ring, we get*dNΓNΓdN *0.

Thus we obtain that*d*0. It contradicts by*d /*0. The theorem is proved.

**4. Generalized Derivations of**

### Γ-Near-Rings

We denote a generalized derivation*f* :*N* → *N*determined by a derivation*d*of*N*byf, d.

We assume that*d*is a nonzero derivation of*N*unless stated otherwise.

* Theorem 4.1. Let*f, d

*be a generalized derivation ofN. Iffx, y*

_{α}*0 for allx, y*∈

*N, α*∈Γ

*,*

*Proof. Assume thatf*x, y* _{α}* 0 for all

*x, y*∈

*N,α*∈Γ. Substitute

*xβy*instead of

*y, obtaining*

*fx, xβy _{α}fxβx, y_{α}dxβx, y_{α}xβfx, y_{α}*0. 4.1

Since the second term is zero, it is clear that

*dxαxβydxαyβx* ∀x, y∈*N, α, β*∈Γ. 4.2

Replacing*y*by*yδz*in4.2and using this equation, we get

*dxαyβx, zδ*0 ∀x, y, z∈*N, α, β, δ*∈Γ. 4.3

Hence either*x*∈*ZN*or*dx * 0. Let*L* {x ∈*N* |*dx * 0}. Then*ZN*and*L*are two

additive subgroups ofN, *ZN*∪*L. However, a group cannot be the union of proper*

subgroups, hence either*NZN*or*NL. Sinced /*0, we are forced to conclude that*N*is

a commutativeΓ-ring.

* Theorem 4.2. Let*f, d

*be a generalized derivation ofN. Iffx, y*±x, y

_{α}

_{α}*for allx, y*∈

*N,*

*α*∈Γ*, thenNis a commutative*Γ*-ring.*

*Proof. Assume thatfx, y _{α}* ±x, y

*for all*

_{α}*x, y*∈

*N,α*∈Γ. Replacing

*y*by

*xβy,β*∈Γ, in the hypothesis, we have

*fx, xβy _{α}*±

*xαxβy*−

*xαyβx*±xβ

*x, y*4.4

_{α}.On the other hand,

*fx, xβy _{α}fxβx, y_{α}dxβx, y_{α}xβfx, y_{α}dxβx, y_{α}xβ*±

*x, y*

_{α}.4.5

It follows from the two expressions for*fx, xβy _{α}*that

*dxαxβydxαyβx* ∀x, y∈*N, α, β*∈Γ. 4.6

Using the same argument as in the proof ofTheorem 4.1, we get that *N* is a commutative

Γ-ring.

* Theorem 4.3. Let*f, d

*be a nonzero generalized derivation ofN. If f acts as a homomorphism on*

*N, thenfis the identity map.*

*Proof. Assume thatf*acts as a homomorphism on*N. Then one obtains*

Replacing*y*by*yβz*in4.7, we arrive at

*fxαfyβzdxαyβzxαfyβz.* 4.8

Sincef, dis a generalized derivation and*f*acts as a homomorphism on*N, we deduce that*

*fxαyβf*z *dxαyβzxαdyβzxαyβfz.* 4.9

ByLemma 3.9ii, we get

*dxαyβf*z *xαfyβfz dxαyβzxαdyβzxαyβf*z, 4.10

and so

*dxαyβfz xαfyβzdxαyβzxαdyβzxαyβfz.* 4.11

That is,

*dxαyβfz xαdyβzxαyβf*z *dxαyβzxαdyβzxαyβfz.* 4.12

Hence, we deduce that

*dxαyβfz*−*z*0 ∀x, y, z∈*N, α, β*∈Γ. 4.13

Because*N*is prime and*d /*0, we have*fz z*for all*z*∈*N. Thus,f*is the identity map.

* Theorem 4.4. Let*f, d

*be a nonzero generalized derivation ofN. If f acts as an antihomomorphism*

*onN, thenfis the identity map.*

*Proof. By the hypothesis, we have*

*fxαyfyαf*x *dxαyxαfy* ∀x, y∈*N, α*∈Γ. 4.14

Replacing*y*by*xβy*in the last equation, we obtain

*fxβyαf*x *dxβxαyxβfxαy.* 4.15

Sincef, dis a generalized derivation and*f*acts as an antihomomorphism on*N, we get*

*dxβyxβfyαf*x *dxαxβyxαfyβfx.* 4.16

ByLemma 3.9ii, we conclude that

and so

*dxαyβf*x *dxαxβy* ∀x, y∈*N, α, β*∈Γ. 4.18

Replacing*y*by*yδz*and using this equation, we have

*dxαyβf*x, z* _{α}*0 ∀x, z∈

*N, α, β*∈Γ. 4.19

Hence we obtain the following alternatives:*dx * 0 or*fx* ∈ *ZN, for allx* ∈ *N. By a*

standard argument, one of these must hold for all*x*∈*N. Sinced /*0, the second possibility

gives that*N*is commutativeΓ-ring byTheorem 3.12, and so we deduce that*f*is the identity

map byTheorem 4.3.

* Theorem 4.5. Let*f, d

*be a generalized derivation ofN*

*such that*

*dZN/0, anda*∈

*N. If*

fx, a* _{α}0 for allx*∈

*N,α, β*∈Γ

*, thena*∈

*ZN.*

*Proof. Since* *dZN/*0, there exists *c* ∈ *ZN* such that *dc/*0. Furthermore, as *d* is a

derivation, it is clear that*dc* ∈ *ZN. Replacingx*by*cβx,β* ∈ Γ, in the hypothesis and

usingLemma 3.9ii, we have

*fcβxαaaαfcβx,*

*dcαxβacαfxβaaαdcβxaαcβf*x. 4.20

Since*c*∈*ZN*and*dc*∈*ZN, we get*

*dcαxβy, a _{α}*0 ∀y∈

*N, α, β, δ*∈Γ. 4.21

By the primeness of*N*and 0*/dc*∈*ZN, we obtain thata*∈*ZN.*

* Theorem 4.6. Let* f, d

*be a generalized derivation ofN, anda*∈

*N. If*fx, a

_{α}*0 for all*

*x*∈*N, thenda*∈*ZN.*

*Proof. Ifa*0, then there is nothing to prove. Hence, we assume that*a /*0.

Replacing*x*by*aβx*in the hypothesis, we have

*faβxαaaαfaβx,*

*daαxβaaαf*xβa*aαdaβxaαaβf*x. 4.22

Using*fxαaaαfx, we have*

*daαxβaaαdaβx* ∀x∈*N, α, β*∈Γ. 4.23

Taking*xδy*instead of*x*in the last equation and using this, we conclude that

Since*N*is a primeΓ-near-ring, we have either*da *0 or*a*∈*ZN. If 0/a*∈*ZN, then*

N,is abelian byLemma 3.2ii. Thus

*fxαa faαx*

*fxαaxαda daαxaαfx* 4.25

and so

da, x*α*0 ∀x∈*N, α*∈Γ. 4.26

That is,*da*∈*ZN. Hence in either case we haveda*∈*ZN. This completes the*

proof.

* Theorem 4.7. Let*f, d

*be a generalized derivation ofN. IfN*

*is a 2-torsion free*Γ

*-near-ring and*

*f*2_{N}_{⊂}_{ZN}_{, then}_{N}_{is a commutative}_{Γ}_{-ring.}

*Proof. Suppose thatf*2_{N}_{⊂}_{ZN. Then we get}

*f*2*xαyf*2xαy2fxαd*yxαd*2*y*∈*ZN* ∀x, y∈*N, α*∈Γ. 4.27

In particular,*f*2_{xαc}_{}_{2fxαdc }* _{xαd}*2

_{c}

_{∈}

_{ZN}_{for all}

_{x}_{∈}

_{N,}_{c}_{∈}

_{ZN,}_{α}_{∈}

_{Γ. Since}

the first summand is an element of*ZN, we have*

2fxαdc *xαd*2c∈*ZN* ∀x∈*N, c*∈*ZN, α*∈Γ. 4.28

Taking*fx*instead of*x*in4.28, we obtain that

2f2xαdc *fxαd*2c∈*ZN* ∀x∈*N, c*∈*ZN, α*∈Γ. 4.29

Since*dc*∈*ZN,f*2_{x}_{∈}* _{ZN, and so}_{f}*2

_{xαdc}

_{∈}

_{ZN}_{for all}

_{x}_{∈}

_{N,}_{c}_{∈}

_{ZN,}_{α}_{∈}

_{Γ,}

we conclude*fxαd*2_{c}_{∈}_{ZN}_{for all}_{x}_{∈}_{N,}_{c}_{∈}_{ZN,}_{α}_{∈}_{Γ.}

Since*N* is prime, we get*d*2_{ZN } _{0 or} _{fN}_{⊆} _{ZN. If}_{f}_{N} _{⊆} _{ZN, then}

*N* is a commutativeΓ-ring byLemma 3.8. Hence, we assume*d*2Z 0. By4.28, we get

2fxαdc∈*ZN*for all*x*∈*N,c*∈*ZN,α*∈Γ.

Since*N* is a 2-torsion free near-ring and*dc* ∈*ZN, we obtain that eitherfN* ⊂

*ZN*or*dZN * 0. If*f*N ⊂*ZN, then we are already done. So, we may assume that*

*dZN *0. Then

*fcαx fxαc,*

*fcαxcαdx fxαcxαdc,* 4.30

and so

Now replacing*x*by*fx*in4.31, and using the fact that*f*2_{N}_{⊂}_{ZN, we get}

*fcαf*x *cαdfxf*2xαc ∀x∈*N, c*∈*ZN.* 4.32

That is,

*fcαf*x *cαdfx*∈*Z* ∀x∈*N, c*∈*ZN, α*∈Γ. 4.33

Again taking*fx*instead of*x*in this equation, one can obtain

*fcαf*2_{x }* _{cαd}_{f}*2

_{x}

_{∈}

_{Z}_{∀x}

_{∈}

_{N, c}_{∈}

_{ZN, α}_{∈}

_{Γ.}

_{4.34}

The second term is equal to zero because of*dZ *0. Hence we have*fcαf*2_{x}_{∈}_{ZN}_{for}

all*x*∈*N,c*∈*ZN,α*∈Γ.

Since*f*2_{N}_{⊂}_{ZN}_{by the hypothesis, we get either}* _{f}*2

_{N }

_{0 or}

_{f}_{ZN}

_{⊂}

_{ZN.}If*f*2_{N } _{0, then the theorem holds by}_{Definition 3.7. If}_{f}_{Z} _{⊂} _{ZN, then}_{fxαf}_{c }

*ffcαx*for all*x*∈*N,c*∈*ZN, and so*

*dxαf*c *fcαfx* ∀x∈*N, c*∈*ZN.* 4.35

Using*fc*∈*ZN, we now havefcαdx*−*fx *0 for all*x*∈*N,c*∈*ZN,α*∈Γ. Since

*fZN*⊂*ZN, we have eitherf*ZN 0 or*df. Ifdf, thenf*is a derivation of*N*

and so*N*is commutativeΓ-ring byLemma 3.11.

Now assume that*f*ZN 0. Returning to the equation4.31, we have

*cαdx*−*fx*0 ∀x∈*N, c*∈*ZN, α*∈Γ. 4.36

Since*c* ∈ *ZN, we have either* *d* *f* or *ZN * 0. Clearly, *d* *f* implies the theorem

holds. If*ZN *0, then*f*2_{N }_{0 by the hypothesis, and so}_{N}_{is a commutative}_{Γ-ring by}

Lemma 3.4iv. Hence, the proof is completed.

* Corollary 4.8. LetNbe a 2-torsion free near-ring, and*f, d

*a nonzero generalized derivation ofN.*

*If*fN, fN* _{α}0,α*∈Γ

*, thenNis a commutative*Γ

*-ring.*

* Lemma 4.9. Let*f, d

*and*g, h

*be two generalized derivations ofN. If h is a nonzero derivation on*

*Nandfxαhy *−gxαdy*for allx, y*∈*N, then*N,*is abelian.*

*Proof. Suppose thatfxαhy gxαdy *0 for all*x, y*∈*N,α*∈Γ.

We substitute*yz*for*y, thereby obtaining*

*fxαhyfxαhz gxαdyg*xαdz 0. 4.37

Using the hypothesis, we get

It follows by Lemma 3.10ii that *hy, z * 0 for all *y, z* ∈ *N. For any* *w* ∈ *N, we have*

*hwαy, wαz * *hwαy, z * *hwαy, z wαhy, z * 0 and so*hwαy, z * 0 for all

*w, y, z*∈*N,α*∈Γ.

An appeal toLemma 3.4iiiyields thatN,is abelian.

* Theorem 4.10. Let*f, d

*and*g, h

*be two generalized derivations ofN. IfNis 2-torsion free and*

*fxαhy *−gxαdy*for allx, y*∈*N, α*∈Γ*, thenf* *0 org* *0.*

*Proof. Ifh*0 or*d*0, then the proof of the theorem is obvious. So, we may assume that*h /*0

and*d /*0. Therefore, we know thatN,is abelian byLemma 4.9.

Now suppose that

*f*xαh*ygxαdy*0 ∀x, y∈*N, α*∈Γ. 4.39

Replacing*x*by*uβv*in this equation and using the hypothesis, we get

*fuβvαhyguβvαdy*

*uαf*vβh*yduαvβhyuαgvβdyhuαvβdy*

0,

4.40

and so

*duαvβhy*−huαvβd*y* ∀u, v, y∈*N, α*∈Γ. 4.41

Taking*yδt*instead of*y*in the above relation, we obtain

*duαvβhyδtduαvyβδht *−huαvβd*yδt*−*huαvβyδdt.* 4.42

That is,

*duαvβyδht *−huαvβyδdt ∀u, v, y, t∈*N, α, β, δ*∈Γ. 4.43

Replacing*y*by*hy*in4.43and using this relation, we have

*huαNβdyδht*−*hyαdt*0 ∀u, y, t∈*N.* 4.44

Since*N*is a primeΓ-near-ring and*h /*0, we obtain that

*dyαht hyαdt,* ∀y, t∈*N.* 4.45

Now again taking*uλv*instead of*x*in the initial hypothesis, we get

Using4.45yields that

*fuαvβhy*2uαhvβd*yguαvβdy*0 ∀u, v, y∈*N,* 4.47

Taking*hv*instead of*v*in this equation, we arrive at

*fuαhvβhy*2uαh2vβd*yguαhvβdy*0. 4.48

By the hypothesis and4.45, we have

0−guαdvβh*y*2uαh2vβd*yguαhvβdy*

−guαhvβd*y*2uαh2vβd*yguαhvβdy,* 4.49

and so

2uαh2vβd*y*0 ∀u, v, y∈*N, α, β*∈Γ. 4.50

Since*N*is a 2-torsion free primeΓ-near-ring, we obtain that*h*2NΓdN 0. An appeal to

Lemma 3.4iiiandivgives that*h*0. This contradicts by our assumption. Thus the proof

is completed.

* Theorem 4.11. Let* f, d

*and*g, h

*be two generalized derivations of*

*N. If*fg, dh

*acts as a*

*generalized derivation onN, thenf* *0 org0.*

*Proof. By calculatingfgxαy*in two diﬀerent ways, we see that*gxαdy fxαhy *0

for all*x, y*∈*N,α*∈Γ. The proof is completed by usingTheorem 4.10.

**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 *H. E. Bell and G. Mason, “On derivations in near-rings,” in Near-Rings and Near-Fields (T ¨ubingen, 1985),*
*vol. 137 of North-Holland Mathematics Studies, pp. 31–35, North-Holland, Amsterdam, The Netherlands,*
1987.

2 M. As¸ci, “Γ-*σ, τderivation on gamma near rings,” International Mathematical Forum. Journal for Theory*
*and Applications, vol. 2, no. 1–4, pp. 97–102, 2007.*

3 *Y. U. Cho, “A study on derivations in near-rings,” Pusan Kyongnam Mathematical Journal, vol. 12, no. 1,*
pp. 63–69, 1996.

4 M. Kazaz and A. Alkan, “Two-sidedΓ-*α*-derivations in prime and semiprimeΓ*-near-rings,” Korean*
*Mathematical Society, vol. 23, no. 4, pp. 469–477, 2008.*

5 M. Uc¸kun, M. A. ¨*Ozt ¨urk, and Y. B. Jun, “On prime gamma-near-rings with derivations,” Korean *
*Math-ematical Society, vol. 19, no. 3, pp. 427–433, 2004.*

### Submit your manuscripts at

### http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

### Mathematics

Journal ofHindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations International Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematical PhysicsAdvances in

### Complex Analysis

Journal ofHindawi Publishing Corporation

http://www.hindawi.com Volume 2014

### Optimization

Journal of Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

### Combinatorics

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

International Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

### Function Spaces

Abstract and Applied Analysis Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

**The Scientific **

**World Journal**

Hindawi Publishing Corporation
http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

## Discrete Mathematics

Journal ofHindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014