On Prime Near Rings with Generalized Derivation

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Volume 2008, Article ID 490316,5pages doi:10.1155/2008/490316

Research Article

On Prime Near-Rings with Generalized Derivation

Howard E. Bell

Department of Mathematics, Faculty of Mathematics and Science, Brock University, St. Catharines, Ontario, Canada L2S 3A1

Correspondence should be addressed to Howard E. Bell,hbell@brocku.ca

Received 29 November 2007; Accepted 25 February 2008

Recommended by Francois Goichot

LetN be a 3-prime 2-torsion-free zero-symmetric left near-ring with multiplicative centerZ. We prove that if N admits a nonzero generalized derivation f such thatfN ⊆ Z, thenN is a commutative ring. We also discuss some related properties.

Copyrightq2008 Howard E. Bell. 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

LetNbe a zero-symmetric left near-ring, not necessarily with a multiplicative identity element; and letZbe its multiplicative center. DefineNto be 3-prime if for alla, bN\{0},aNb /{0}; and callN2-torsion-free if N, has no elements of order 2. A derivation onNis an additive endomorphism Dof N such thatDxy xDy Dxyfor allx, yN. A generalized derivationfwith associated derivationDis an additive endomorphismf:NNsuch that

fxy fxyxDyfor allx, yN. In the case of rings, generalized derivations have received significant attention in recent years.

In1, we proved the following.

Theorem A. If N is 3-prime and 2-torsion-free and D is a derivation such thatD20, thenD0.

Theorem B. If N is a 3-prime 2-torsion-free near-ring which admits a nonzero derivation D for which

DNZ, then N is a commutative ring.

Theorem C. If N is a 3-prime 2-torsion-free near-ring admitting a nonzero derivation D such that

DxDy DyDxfor allx, yN, then N is a commutative ring.

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We will need three easy lemmas.

Lemma 1.1see1, Lemma 3. Let N be a 3-prime near-ring.

iIfzZ\ {0}, then z is not a zero divisor.

iiIfZ\ {0}contains an element z such thatzzZ, then (N,) is abelian.

iiiIf D is a nonzero derivation andxNis such thatxDN {0}orDNx {0}, then

x0.

Lemma 1.2see2, Proposition 1. If N is an arbitrary near-ring and D is a derivation on N, then D(xy) = D(x)y + xD(y) for allx, yN.

Lemma 1.3. Let N be an arbitrary near-ring and let f be a generalized derivation on N with associated

derivation D. Then

fabaDbcfabcaDbca, b, cN. 1.1

Proof. Clearlyfabc fabcabDc fabaDbcabDc; and by usingLemma 1.2, we obtainfabc fabcaDbc fabcaDbcabDc.

Comparing these two expressions forfabcgives the desired conclusion.

2. The main theorem

Our best result is an extension of Theorem B.

Theorem 2.1. Let N be a 3-prime 2-torsion-free near-ring. If N admits a nonzero generalized derivation

f such thatfNZ, then N is a commutative ring.

In the proof of this theorem, as well as in a later proof, we make use of a further lemma.

Lemma 2.2. Let R be a 3-prime near-ring, and let f be a generalized derivation with associated

derivationD /0. IfDfN {0}, thenfDN {0}.

Proof. We are assuming thatDfx 0 for allxN. It follows thatDfxy Dfxy

DxDy 0 for allx, yN, that is,

fxDy DxDy xD2y 0 x, yN. 2.1

ApplyingDagain, we get

fxD2y D2xDy DxD2y DxD2y xD3y 0 x, yN. 2.2

TakingDyinstead ofyin2.1givesfxD2yDxD2yxD3y 0, hence2.2yields

D2xDy DxD2y 0 x, yN. 2.3

Now, substituteDxforxin2.1, obtainingfDxDy D2xDy DxD2y 0;

and use 2.3to conclude that fDxDy 0 for all x, yN. Thus, by Lemma 1.1iii,

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Proof ofTheorem 2.1. Sincef /0, there existsxNsuch that 0 /fxZ. Sincefx fx

fxxZ, N,is abelian by Lemma 1.1ii. To complete the proof, we show thatNis multiplicatively commutative.

First, consider the caseD0, so thatfxy fxyZfor allx, yN. Thenfxyw

wfxy, hencefxywwy 0 for allx, y, wN. Choosingx such thatfx/0 and invokingLemma 1.1i, we getywwy0 for ally, wN.

Now assume thatD /0, and letcZ\ {0}. Thenfxc fxcxDcZ; therefore,

fxcxDcyyfxcxDcfor allx, yN, and byLemma 1.3, we see thatfxcy

xDcyyfxcyxDc. Since bothfxandDcare inZ, we haveDcxyyx 0 for allx, yN, and provided thatDZ/{0}, we can conclude thatNis commutative.

Assume now thatD /0 andDZ {0}. In particular,Dfx 0 for allxN. Note that forcNsuch thatfc 0,fcx cDxZ; hence byLemma 2.2,DxDyZand

DyDxZfor eachx, yN. If one of these is 0, the other is a central element squaring to 0, hence is also 0. The remaining possibility is thatDxDyandDyDxare nonzero central elements, in which case Dxis not a zero divisor. Thus DxDxDy DxDyDx yields DxDxDyDyDx 0 DxDyDyDx. Consequently, N is commutative by Theorem C.

3. On Theorems A and C

Theorem C does not extend to generalized derivations, even ifNis a ring. As in3, consider the ringH of real quaternions, and definef : HH byfx ixxi. It is easy to check thatfis a generalized derivation with associated derivation given byDx xiix, and that

fxfy fyfxfor allx, yH.

Theorem A also does not extend to generalized derivations, as we see by lettingNbe the ringM2Fof 2×2 matrices over a fieldFand lettingfbe defined byfx e12x. However,

we do have the following results.

Theorem 3.1. Let N be a 3-prime near-ring, and let f be a generalized derivation on N with associated

derivation D. Iff20, thenD30. Moreover, if N is 2-torsion-free, thenDZ {0}.

Proof. We have

f2xy ffxyxDy fxDy fxDy xD2y 0 x, yN. 3.1

Applyingfto3.1gives

fxD2y fxD2y fxD2y xD3y 0 x, yN. 3.2

SubstitutingDyforyin3.1gives

fxD2y fxD2y xD3y 0; 3.3

Therefore, by3.2and3.3,

fxD2y 0 x, yN. 3.4

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Suppose now thatNis 2-torsion-free and that DZ/{0}, and letzZbe such that

Dz/0. Then ifx, yNandfNx{0}, thenfyzxfyzxyDzx0yDzx; and sinceNis 3-prime andDzis not a zero divisor,x0. It now follows from3.4that

D2 0 and hence by Theorem A that D 0. But this contradicts our assumption that

DZ/{0}, henceDZ {0}as claimed.

Theorem 3.2. Let N be a 3-prime and 2-torsion-free near-ring with 1. If f is a generalized derivation on

N such thatf20 andf1Z, thenf 0.

Proof. Note thatfx f1x f1x1Dx, so

fx cxDx, cZ. 3.5

Ifc0, thenfDandD20, soD0 by Theorem A and thereforef0.

Ifc /0, thencis not a zero divisor, hence by3.4D20 andD0. But thenfx cx

and f2x c2x 0 for all x N. Since c2 is not a zero divisor, we get N {0}—a

contradiction. Thus,c0 and we are finished.

4. More on Theorem C

In4, the author studied generalized derivationsfwith associated derivationDwhich have the additional property that

fxy Dxyxfyx, yN.

Our final theorem, a weak generalization of Theorem C, was stated in4; but the proof given was not correct.At one point, both left and right distributivity were assumed.We now have all the results required for a proof.

Theorem 4.1. Let N be a 3-prime 2-torsion-free near-ring which admits a generalized derivationfwith nonzero associated derivation D such that f satisfies. Iffxfy fyfxfor allx, yN, then N is a commutative ring.

Proof. It is correctly shown in4that N, is abelian and eitherfNZorDfN {0}. Hence, in view ofTheorem 2.1, we may assume thatDfN 0 and therefore, byLemma 2.2, thatfDN {0}. We calculatefDxDyin two ways. Using the defining property off, we obtainfDxDy fDxDy DxD2y DxD2y; and using, we obtain

fDxDy D2xDyDxfDy D2xDy. Thus,D2xDy DxD2yfor all

x, yN. But sinceDfN {0},2.3holds in this case as well; thereforeD2xDy 0 for

allx, yN, hence byLemma 1.1iiiD20. Thus,D0, contrary to our original hypothesis,

so that the caseDfN {0}does not in fact occur.

Acknowledgment

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References

1H. E. Bell and G. Mason, “On derivations in near-rings,” in Near-Rings and Near-Fields (T ¨ubingen, 1985), G. Betsch, Ed., vol. 137 of North-Holland Mathematics Studies, pp. 31–35, North-Holland, Amsterdam, The Netherlands, 1987.

2X.-K. Wang, “Derivations in prime near-rings,” Proceedings of the American Mathematical Society, vol. 121, no. 2, pp. 361–366, 1994.

3H. E. Bell and N.-U. Rehman, “Generalized derivations with commutativity and anti-commutativity conditions,” Mathematical Journal of Okayama University, vol. 49, no. 1, pp. 139–147, 2007.

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