On Prime Gamma Near Rings with Generalized Derivations

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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,

1

and Isamiddin S. Rakhimov

2

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

2Department 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.

LetN be a 2-torsion free prime Γ-near-ring with center ZN. Let f, d and g, h be two generalized derivations onN. We prove the following results:iiffx, yα 0 orfx, yα ±x, yα orf2xZNfor allx, y N,α Γ, thenNis a commutativeΓ-ring.iiIfa N

andfx, aα 0 for allxN,α∈Γ, thendaZN.iiiIffg, dhacts as a generalized derivation onN, thenf0 org0.

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. IfaN, andfx, aα0 for allxN, α∈Γ, thendis central. Also we prove that if

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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 onNsuch that for eachα∈Γ,N,, αis

a left near-ring;

iiixαyβz xαyβz, for allx, y, zNandα, β∈Γ.

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

N0{x∈N: 0αx0, α∈Γ}is called the zero-symmetric part ofN. AΓ-near-ringNis said

to be zero-symmetric ifNN0. Throughout this paper,Nwill denote a zero symmetric left

Γ-near-ring with multiplicative centreZN. Recall that aΓ-near-ringNis prime ifxΓNΓy

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

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

1, thatdxαy dxαyxαdyfor allx, yN, α ∈ Γ. Further, an elementxN for

whichdx 0 is called a constant. Forx, yN, α ∈ Γ, the symbolx, yα will denote the

commutatorxαyyαx, while the symbolx, ywill denote the additive-group commutator

xyxy. An additive mappingf :NNis called a generalized derivation if there exits

a derivationdofNsuch thatfxαy fxαyxαdyfor allx, yN, 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 aΓ-near-ringN. ThenN satisfies the following

partial distributive law:aαdb daαbβcaαdbβcdaαbβcanddaαbaαdbβc

daαbβcaαdbβcfor alla, b, cN, α, β∈Γ.

Proof. For all a, b, cN, α, β ∈ Γ, we getdaαbβc 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 aΓ-near-ringNand supposeuNis not a left zero divisor. If

u, duα0, α∈Γ, thenx, uis a constant for everyxN.

Proof. Fromuαux uαuuαx, for allxN,α∈Γ, we obtainuαdux duαux

uαdu duαuuαdx duαx, which reducesuαdx duαuduαuuαdx, for

allα∈Γ.

Sinceduαuuαdu,α∈Γ, this equation is expressible asuαdx dudx

du 0uαdx, u. Thusdx, u 0.

Theorem 3.3. LetNbe aΓ-near-ring having no nonzero divisors of zero. IfN admits a nontrivial

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Proof. Letcbe any additive commutator. Thencis a constant byLemma 3.2. Moreover, for

anywN,α∈Γ,wαcis an additive commutator, hence also a constant. Thus, 0dwαc

wαdc dwαcanddwαc 0, for allα∈ Γ. Sincedw/0 for allwN, we conclude

thatc0.

Lemma 3.4. LetNbe a primeΓ-near-ring.

iIfzZN− {0}, thenzis not a zero divisor inN.

iiIfZN− {0}contains an elementzfor whichzzZN, thenN,is abelian.

iiiLetdbe a nonzero derivation onN. ThenxΓdN {0}impliesx0, anddNΓx{0}

impliesx0.

ivIfNis 2-torsion free anddis a derivation onNsuch thatd2 0, thend0.

Proof. iIfzZN− {0}andzαx0,xN,α∈Γ, thenzαrβx0,x,rN,α∈Γ. Thus

we getzΓNΓx0, by primeness ofN,x0.

iiLetzZN− {0}be an element such thatzzZN, and letx, yN, α∈Γ.

Sincezzis distributive, we getx yαz z xαz z yαz z xαz xαz

yαz yαz zαx x y y.

On the other hand,xyαzz xyαz xyαzzαxyxy. Thus,

xxyyxyxyand thereforexyyx. HenceN,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, yN, α ∈ Γ, we have 0 d2xαy dxαdy dxαy

xαd2y dxαdy dxαdy d2xαy 2dxαdy. Since N is 2-torsion free,

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

da 0. Thusd0.

Theorem 3.5. If a primeΓ-near-ringNadmits a nontrivial derivationdfor whichdNZN,

thenN,is abelian. Moreover, ifNis 2-torsion free, thenNis a commutativeΓ-ring.

Proof. Letc be an arbitrary constant, and letx be anon-constant. Thendxαc xαdc

dxαc dxαcZN,α ∈Γ. SincedxZN− {0}, it follows easily thatcZN.

Sinceccis a constant for all constantsc, it follows fromLemma 3.4iithatN,is abelian,

provided that there exists a nonzero constant.

Assume, then, that 0 is the only constant. Sincedis obviously commuting, it follows

fromLemma 3.2that alluwhich are not zero divisors belong to the center ofN,, denoted

byZN. In particular, ifx /0,dxZN. But then for allyN,dydx−dy−dx

dy, x 0, hencey, x 0.

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

aαdbβcdaαbβcfor alla, b, cN, α, β ∈ Γ, and using the fact that daαbZN,

α∈ Γ, we getcαaβdb cαdaβb aαdbβcaαdbβc, α, β ∈ Γ. SinceN,is abelian

anddNZN, this equation can be rearranged to yielddbαc, aβdaαb, cβfor all

a, b, cN, α, β∈Γ.

Suppose now thatN is not commutative. Choosingb, cN, withb, cβ/0,β ∈ Γ,

and lettinga dx, we getd2xαb, c

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elementsd2xcannot be a nonzero divisor of zero, we conclude thatd2x 0 for allxN.

But byLemma 3.4iv, this cannot happen for nontriviald.

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

dx, dyα 0 for all x, yN, α ∈ Γ. Then N, is abelian. Moreover, ifN is 2-torsion

free, thenNis a commutativeΓ-ring.

Proof. ByLemma 3.4ii, if bothzandzzcommute element-wise withdN, thenzαdc

0, α ∈Γ, for all additive commutatorsc. Thus, takingz dx, we getdxαdc 0 for all

xN, α ∈Γ, sodc 0 byLemma 3.4iii. Sincewαcis also an additive commutator for

anywN,α∈Γ, we havedwαc 0 dwαc, and another application ofLemma 3.4iii

givesc0.

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

ddxαyβdz dxαdyβdz d2xαyβdz for all x, y, z N, α, β Γ, hence,

d2xαyβdz ddxαyβdzdxαdyβdz dxαdyβdz dzαddxβy

dxβdy dzαd2xβyd2xαdzβy, α, βΓ. Thusd2xαyβdzdzβy 0 for

allx, y zN,α, β∈Γ.

Replacingyδt, δ∈Γ, we obtaind2xαyδtβdz d2xαdzβyδtd2xαyβdzδt,

for allx, y, z, tN,α, β, δ∈Γ, so thatd2xαyβt, dzδ0 for allx, y, z, tN,α, β, δ∈Γ.

The primeness ofNshows that eitherd2 0 ordN ZN, and since the first of these

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

Γ-ring byTheorem 3.5.

Definition 3.7. LetNbe aΓ-near-ring andda derivation ofN. An additive mappingf:N

Nis said to be a right generalized derivation ofNassociated withdif

fxαyfxαyxαdy ∀x, y∈R, α∈Γ, 3.1

andfis said to be a left generalized derivation ofNassociated withdif

fxαydxαyxαfy ∀x, y∈R, α∈Γ. 3.2

Finally,f is said to be a generalized derivation ofNassociated withdif it is both a left and

right generalized derivation ofNassociated withd.

Lemma 3.8. Letfbe a right generalized derivation of aΓ-near ringNassociated withd. Then

ifxαy xαdy fxαyfor allx, yN, α∈Γ;

iifxαy xαfy dxαyfor allx, yN, α∈Γ.

Proof. iFor anyx, yN, α∈Γ, we get

fxαyyfyyxαdyyfxαyfxαyxαdyxαdy,

fxαyxαyfxαyxαdyfxαyxαdy.

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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 aΓ-near ringNassociated withd. Then

i fxαyxαdyβzfxαyβzxαdyβz, for allx, yN,α, β∈Γ.

ii dxαyxαfyβzdxαyβzxαfyβz, for allx, yN,α, β∈Γ.

Proof. iFor anyx, y, zN,α, β∈Γ, we getfxαyβz fxαyβzxαyβdz. On the other hand,

fxαyβzfxαyβzxαdyβzfxαyβzxαdyβzxαyβdz. 3.6

From these two expressions offxαyβz, we obtain that, for allx, y, zN,α, β∈Γ,

fxαyxαdyβzfxαyβzxαdyβz. 3.7

iiThe proof is similar.

Lemma 3.10. LetNbe a primeΓ-near-ring,fa nonzero generalized derivation ofNassociated with

the nonzero derivationdandaN.iIfaΓfN 0, thena0.iiIffNΓa0, thena0.

Proof. iFor anyx, yN, α, β∈Γ, we get 0aβfxαy aβfxαyaβxαdy aβxαdy.

HenceaΓNΓdN 0. SinceNis a primeΓ-near-ring andd /0, we obtaina0.

iiA similar argument works iffNΓa0.

Lemma 3.11. LetNbe a primeΓ-near-ring. Letfbe a generalized derivation ofNassociated with

the nonzero derivationd. IfNis a 2-torsion freeΓ-near-ring andf20, thenf0.

Proof. iFor anyx, yN,α∈Γ, we get

0f2xαyffxαyffxαyxαdyf2xαy2fxαdyxαd2y.

3.8

By the hypothesis,

2fxαdyxαd2y0 ∀x, yN, αΓ. 3.9

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ByLemma 3.9ii, we obtain thatd2N 0 or f 0. Ifd2N 0 thend 0 from

Lemma 3.4iv, a contradiction. So we findf0.

Theorem 3.12. LetNbe a primeΓ-near-ring with a nonzero generalized derivationfassociated with

d. IffNZN, thenN,is abelian. Moreover, ifNis 2-torsion free, thenNis commutative

Γ-ring.

Proof. Suppose that aN, such thatfa/0. So,fa ∈ ZN− {0} and fa fa

ZN− {0}. For allx, yN,α∈Γ, we havexyαfafa fafaαxy.

That is,xαfa xαfa yαfa yαfa faαxfaαxfaαyfaαy, for

allx, yN,α∈Γ.

SincefaZN, we getfaαxfaαyfaαyfaαx, and so,faαx, y 0

for allx, yN, α∈Γ.

SincefaZN− {0}andNis a primeΓ-near-ring, it follows thatx, y 0, for all

x, yN. ThusN,is abelian.

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

Lemma 3.4ii, we havezαdxβyzαxfy dxαyβzxαfyβz. UsingfNZN

andN,being abelian, we obtain that

zαdxβydxαyβz x, zαβf

y, ∀x, y∈N, α, β∈Γ. 3.10

Substitutingfzforzin3.10, we getfzβdx, yα0 for allx, yN, α, β∈Γ.

Sincefz∈ZNand f a nonzero generalized derivation with associated withd, we

getdNZN. So,Nis a commutativeΓ-ring byTheorem 3.3.

Theorem 3.13. LetNbe a primeΓ-near-ring with a nonzero generalized derivationfassociated with

d. IffN, fNα 0,α∈Γ, thenN,is abelian. Moreover, ifNis 2-torsion free, thenNis

commutativeΓ-ring.

Proof. By the same argument as inTheorem 3.12, it is shown that if bothzandzzcommute

elementwise withfN, then we have

zαfx, y0 ∀x, y∈N, α∈Γ. 3.11

Substitutingft, t ∈ N forz in 3.11, we getftαfx, y 0, α ∈ Γ. ByLemma 3.9i,

we obtain that fx, y 0 for allx, yN,α ∈ Γ. For anywN, β ∈ Γ, we have 0

fwβx, wβy fwβx, y dwβx, ywβfx, yand so, we obtaindwβx, y 0, for

anywN,β∈Γ. FromLemma 3.4iii, we getx, y 0 for anyx, yN.

Now we assume thatNis 2-torsion free. By the assumptionfN, fNα0,α∈Γ,

we have

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

Hence we get

fzαdfxβyfzαfxβfydfxαyβfz fxαfyβfz,

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and so,

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

If we takeyδwinstead ofyin3.14, then

dfxαyδwβfz fzαdfxβyδwdfxαyδfzβw

∀x, y, z∈N, α, β, δ∈Γ, 3.15

and so,

dfxαyδwβfz−dfxαyδfzβwdfxαyδfz, wδ0

∀x, y, z∈N, α, β, δ∈Γ. 3.16

Thus we getdfxΓNΓfz, wδ 0, for allx, y, zN,α, β, δ ∈ Γ. SinceN is a prime

Γ-near-ring, we havedfN 0 orfNZN. Let us assume thatdfN 0. Then

0dfxαyddxαyxαfy 3.17

and so,

d2xαydxαdydxαfy0, ∀x, y∈N, α∈Γ. 3.18

Replacingybyyβz,β∈Γ, in3.18, we get

0d2xαyβzdxαdyβzdxαfyβz

d2xαyβzdxαdyβzdxαyβdz dxαfyβzdxαyβdz

d2xαydxαdydxαfyβz2dxαyβdz ∀x, y, z∈N, α, β∈Γ.

3.19

Using3.18andNbeing 2-torsion freeΓ-near-ring, we getdNΓNΓdN 0.

Thus we obtain thatd0. It contradicts byd /0. The theorem is proved.

4. Generalized Derivations of

Γ-Near-Rings

We denote a generalized derivationf :NNdetermined by a derivationdofNbyf, d.

We assume thatdis a nonzero derivation ofNunless stated otherwise.

Theorem 4.1. Letf, dbe a generalized derivation ofN. Iffx, yα 0 for allx, yN, α∈Γ,

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Proof. Assume thatfx, yα 0 for allx, yN,α∈Γ. Substitutexβyinstead ofy, obtaining

fx, xβyαfxβx, yαdxβx, yαxβfx, yα0. 4.1

Since the second term is zero, it is clear that

dxαxβydxαyβx ∀x, y∈N, α, β∈Γ. 4.2

Replacingybyyδzin4.2and using this equation, we get

dxαyβx, zδ0 ∀x, y, z∈N, α, β, δ∈Γ. 4.3

Hence eitherxZNordx 0. LetL {x ∈N |dx 0}. ThenZNandLare two

additive subgroups ofN, ZNL. However, a group cannot be the union of proper

subgroups, hence eitherNZNorNL. Sinced /0, we are forced to conclude thatNis

a commutativeΓ-ring.

Theorem 4.2. Letf, dbe a generalized derivation ofN. Iffx, yα ±x, yα for allx, yN,

α∈Γ, thenNis a commutativeΓ-ring.

Proof. Assume thatfx, yα ±x, yαfor allx, yN,α∈Γ. Replacingybyxβy,β∈Γ, in the hypothesis, we have

fx, xβyα±xαxβyxαyβx±xβx, yα. 4.4

On the other hand,

fx, xβyαfxβx, yαdxβx, yαxβfx, yαdxβx, yα±x, yα.

4.5

It follows from the two expressions forfx, xβyαthat

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. Letf, dbe a nonzero generalized derivation ofN. If f acts as a homomorphism on

N, thenfis the identity map.

Proof. Assume thatfacts as a homomorphism onN. Then one obtains

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Replacingybyyβzin4.7, we arrive at

fxαfyβzdxαyβzxαfyβz. 4.8

Sincef, dis a generalized derivation andfacts as a homomorphism onN, we deduce that

fxαyβfz dxαyβzxαdyβzxαyβfz. 4.9

ByLemma 3.9ii, we get

dxαyβfz xαfyβfz dxαyβzxαdyβzxαyβfz, 4.10

and so

dxαyβfz xαfyβzdxαyβzxαdyβzxαyβfz. 4.11

That is,

dxαyβfz xαdyβzxαyβfz dxαyβzxαdyβzxαyβfz. 4.12

Hence, we deduce that

dxαyβfzz0 ∀x, y, z∈N, α, β∈Γ. 4.13

BecauseNis prime andd /0, we havefz zfor allzN. Thus,fis the identity map.

Theorem 4.4. Letf, dbe a nonzero generalized derivation ofN. If f acts as an antihomomorphism

onN, thenfis the identity map.

Proof. By the hypothesis, we have

fxαyfyαfx dxαyxαfy ∀x, y∈N, α∈Γ. 4.14

Replacingybyxβyin the last equation, we obtain

fxβyαfx dxβxαyxβfxαy. 4.15

Sincef, dis a generalized derivation andfacts as an antihomomorphism onN, we get

dxβyxβfyαfx dxαxβyxαfyβfx. 4.16

ByLemma 3.9ii, we conclude that

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and so

dxαyβfx dxαxβy ∀x, y∈N, α, β∈Γ. 4.18

Replacingybyyδzand using this equation, we have

dxαyβfx, zα0 ∀x, z∈N, α, β∈Γ. 4.19

Hence we obtain the following alternatives:dx 0 orfxZN, for allxN. By a

standard argument, one of these must hold for allxN. Sinced /0, the second possibility

gives thatNis commutativeΓ-ring byTheorem 3.12, and so we deduce thatfis the identity

map byTheorem 4.3.

Theorem 4.5. Letf, d be a generalized derivation ofN such that dZN/0, andaN. If

fx, aα0 for allxN,α, β∈Γ, thenaZN.

Proof. Since dZN/0, there exists cZN such that dc/0. Furthermore, as d is a

derivation, it is clear thatdcZN. Replacingxbycβx,β ∈ Γ, in the hypothesis and

usingLemma 3.9ii, we have

fcβxαaaαfcβx,

dcαxβacαfxβaaαdcβxaαcβfx. 4.20

SincecZNanddcZN, we get

dcαxβy, aα0 ∀y∈N, α, β, δ∈Γ. 4.21

By the primeness ofNand 0/dcZN, we obtain thataZN.

Theorem 4.6. Let f, d be a generalized derivation ofN, andaN. If fx, aα 0 for all

xN, thendaZN.

Proof. Ifa0, then there is nothing to prove. Hence, we assume thata /0.

Replacingxbyaβxin the hypothesis, we have

faβxαaaαfaβx,

daαxβaaαfxβaaαdaβxaαaβfx. 4.22

Usingfxαaaαfx, we have

daαxβaaαdaβx ∀x∈N, α, β∈Γ. 4.23

Takingxδyinstead ofxin the last equation and using this, we conclude that

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SinceNis a primeΓ-near-ring, we have eitherda 0 oraZN. If 0/aZN, then

N,is abelian byLemma 3.2ii. Thus

fxαa faαx

fxαaxαda daαxaαfx 4.25

and so

da, xα0 ∀x∈N, α∈Γ. 4.26

That is,daZN. Hence in either case we havedaZN. This completes the

proof.

Theorem 4.7. Letf, dbe a generalized derivation ofN. IfN is a 2-torsion freeΓ-near-ring and

f2NZN, thenNis a commutativeΓ-ring.

Proof. Suppose thatf2NZN. Then we get

f2xαyf2xαy2fxαdyxαd2yZN ∀x, y∈N, α∈Γ. 4.27

In particular,f2xαc2fxαdc xαd2c ZNfor allxN,cZN,αΓ. Since

the first summand is an element ofZN, we have

2fxαdc xαd2c∈ZN ∀x∈N, cZN, α∈Γ. 4.28

Takingfxinstead ofxin4.28, we obtain that

2f2xαdc fxαd2c∈ZN ∀x∈N, cZN, α∈Γ. 4.29

SincedcZN,f2xZN, and sof2xαdcZNfor allxN,cZN,αΓ,

we concludefxαd2cZNfor allxN,cZN,αΓ.

SinceN is prime, we getd2ZN 0 or fN ZN. IffN ZN, then

N is a commutativeΓ-ring byLemma 3.8. Hence, we assumed2Z 0. By4.28, we get

2fxαdc∈ZNfor allxN,cZN,α∈Γ.

SinceN is a 2-torsion free near-ring anddcZN, we obtain that eitherfN

ZNordZN 0. IffN ⊂ZN, then we are already done. So, we may assume that

dZN 0. Then

fcαx fxαc,

fcαxcαdx fxαcxαdc, 4.30

and so

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Now replacingxbyfxin4.31, and using the fact thatf2NZN, we get

fcαfx cαdfxf2xαc ∀x∈N, cZN. 4.32

That is,

fcαfx cαdfxZ ∀x∈N, cZN, α∈Γ. 4.33

Again takingfxinstead ofxin this equation, one can obtain

fcαf2x cαdf2x Z ∀xN, cZN, αΓ. 4.34

The second term is equal to zero because ofdZ 0. Hence we havefcαf2xZNfor

allxN,cZN,α∈Γ.

Sincef2NZNby the hypothesis, we get eitherf2N 0 orfZNZN.

Iff2N 0, then the theorem holds byDefinition 3.7. IffZ ZN, thenfxαfc

ffcαxfor allxN,cZN, and so

dxαfc fcαfx ∀x∈N, cZN. 4.35

UsingfcZN, we now havefcαdxfx 0 for allxN,cZN,α∈Γ. Since

fZNZN, we have eitherfZN 0 ordf. Ifdf, thenfis a derivation ofN

and soNis commutativeΓ-ring byLemma 3.11.

Now assume thatfZN 0. Returning to the equation4.31, we have

cαdxfx0 ∀x∈N, cZN, α∈Γ. 4.36

SincecZN, we have either d f or ZN 0. Clearly, d f implies the theorem

holds. IfZN 0, thenf2N 0 by the hypothesis, and soNis a commutativeΓ-ring by

Lemma 3.4iv. Hence, the proof is completed.

Corollary 4.8. LetNbe a 2-torsion free near-ring, andf, da nonzero generalized derivation ofN.

IffN, fNα0,α∈Γ, thenNis a commutativeΓ-ring.

Lemma 4.9. Letf, dandg, hbe two generalized derivations ofN. If h is a nonzero derivation on

Nandfxαhy −gxαdyfor allx, yN, thenN,is abelian.

Proof. Suppose thatfxαhy gxαdy 0 for allx, yN,α∈Γ.

We substituteyzfory, thereby obtaining

fxαhyfxαhz gxαdygxαdz 0. 4.37

Using the hypothesis, we get

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It follows by Lemma 3.10ii that hy, z 0 for all y, zN. For any wN, we have

hwαy, wαz hwαy, z hwαy, z wαhy, z 0 and sohwαy, z 0 for all

w, y, zN,α∈Γ.

An appeal toLemma 3.4iiiyields thatN,is abelian.

Theorem 4.10. Letf, dandg, hbe two generalized derivations ofN. IfNis 2-torsion free and

fxαhy −gxαdyfor allx, yN, α∈Γ, thenf 0 org 0.

Proof. Ifh0 ord0, then the proof of the theorem is obvious. So, we may assume thath /0

andd /0. Therefore, we know thatN,is abelian byLemma 4.9.

Now suppose that

fxαhygxαdy0 ∀x, y∈N, α∈Γ. 4.39

Replacingxbyuβvin this equation and using the hypothesis, we get

fuβvαhyguβvαdy

uαfvβhyduαvβhyuαgvβdyhuαvβdy

0,

4.40

and so

duαvβhy−huαvβdy ∀u, v, y∈N, α∈Γ. 4.41

Takingyδtinstead ofyin the above relation, we obtain

duαvβhyδtduαvyβδht −huαvβdyδthuαvβyδdt. 4.42

That is,

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

Replacingybyhyin4.43and using this relation, we have

huαNβdyδhthyαdt0 ∀u, y, t∈N. 4.44

SinceNis a primeΓ-near-ring andh /0, we obtain that

dyαht hyαdt, ∀y, t∈N. 4.45

Now again takinguλvinstead ofxin the initial hypothesis, we get

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Using4.45yields that

fuαvβhy2uαhvβdyguαvβdy0 ∀u, v, y∈N, 4.47

Takinghvinstead ofvin this equation, we arrive at

fuαhvβhy2uαh2vβdyguαhvβdy0. 4.48

By the hypothesis and4.45, we have

0−guαdvβhy2uαh2vβdyguαhvβdy

−guαhvβdy2uαh2vβdyguαhvβdy, 4.49

and so

2uαh2vβdy0 ∀u, v, y∈N, α, β∈Γ. 4.50

SinceNis a 2-torsion free primeΓ-near-ring, we obtain thath2NΓdN 0. An appeal to

Lemma 3.4iiiandivgives thath0. 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 calculatingfgxαyin two different ways, we see thatgxαdy fxαhy 0

for allx, yN,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.

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