International Journal of Mathematics and Mathematical Sciences

VolumeΒ 2008Β (2008), Article IDΒ 490316, 5 pages

http://dx.doi.org/10.1155/2008/490316

## On Prime Near-Rings with Generalized Derivation

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

Received 29 November 2007; Accepted 25 February 2008

Academic Editor: FrancoisΒ Goichot

Copyright Β© 2008 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.

#### Abstract

Let be a 3-prime 2-torsion-free zero-symmetric left near-ring with multiplicative center . We prove that if admits a nonzero generalized derivation such that , then is a commutative ring. We also discuss some related properties.

#### 1. Introduction

Let be a zero-symmetric left near-ring, not necessarily with a multiplicative identity element; and let be its multiplicative center. Define to be 3-prime if for all , ; and call 2-torsion-free if () has no elements of order 2. A derivation on is an additive endomorphism of such that for all . A generalized derivation with associated derivation is an additive endomorphism such that for all . In the case of rings, generalized derivations have received significant attention in recent years.

In [1], we proved the following.

Theorem A. *If N is 3-prime and 2-torsion-free
and D is a derivation such that , then .*

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

In this paper, we investigate possible analogs of these results, where is replaced by a generalized derivation .

We will need three easy lemmas.

Lemma 1.1 (see [1, Lemma 3]). *Let N be a
3-prime near-ring. *

(i)* If, then z is not a zero divisor. *(ii)*If contains an element z such that , then () is abelian. *(iii)*If D is a
nonzero derivation and is such that or , then .*

Lemma 1.2 (see [2, 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 all .*

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

*Proof. *Clearly and by using Lemma 1.2, we obtain .

Comparing these two expressions for gives 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
that , 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 derivation . If , then .*

*Proof. *We are assuming that for all . It follows that for all , that is,

Applying again, we get

Taking instead of in (2.1) gives , hence (2.2) yields

Now, substitute for in (2.1), obtaining ; and use (2.3) to conclude that for all . Thus, by Lemma 1.1(iii), for all .

*Proof of Theorem 2.1. *Since ,
there exists such that. Since is abelian by Lemma 1.1(ii). To complete the proof, we show that is multiplicatively commutative.

First, consider the case =
0, so that for all . Then ,
hence
for all . Choosing such that and invoking Lemma 1.1(i), we get β = 0 for all .

Now
assume that , and
let . Then ;
therefore, for all ,
and by Lemma 1.3, we see that . Since both and are in , we have for all ,
and provided that , we
can conclude that is commutative.

Assume now that and . In particular, for all . Note that for such that , ;
hence by Lemma 2.2, and for each . If one of these is 0, the other is a central
element squaring to 0, hence is also 0. The remaining possibility is that and are nonzero central elements, in which
case is not a zero divisor. Thus yields . Consequently, is commutative by Theorem C.

#### 3. On Theorems A and C

Theorem C does not extend to generalized derivations, even if is a ring. As in [3], consider the ring of real quaternions, and define by . It is easy to check that is a generalized derivation with associated derivation given by , and that for all .

Theorem A also does not extend to generalized derivations, as we see by letting be the ring of 2 Γ 2 matrices over a field and letting be defined by . 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. If , then. Moreover, if N is 2-torsion-free, then .*

*Proof. *We have
Applying to (3.1) gives

Substituting for in (3.1) gives

Therefore,
by (3.2) and (3.3),

It
now follows from (3.3) that for all ;
and since is 3-prime, .

Suppose now that is 2-torsion-free and that ,
and let be such that . Then if and , then ; and since is 3-prime and is not a zero divisor, = 0. It now follows from (3.4)
that and hence by Theorem A that = 0. But this contradicts our assumption that ,
hence as claimed.

Theorem 3.2. *Let N be a 3-prime and 2-torsion-free
near-ring with . If f is a
generalized derivation on N such that and , then .*

*Proof. *Note that , so

If =
0, then and , so =
0 by Theorem A and therefore =
0.

If ,
then is not a zero divisor, hence by (3.4) and = 0. But then and for all . Since is not a zero divisor, we get βa contradiction. Thus, = 0 and we are finished.

#### 4. More on Theorem C

In [4], the author studied generalized derivations with associated derivation which have the additional property that

Our final theorem, a weak generalization of Theorem C, was stated in [4]; 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 derivation with nonzero associated derivation D such
that f satisfies (β). If for all , then N is a commutative ring.*

*Proof. *It is correctly shown in [4] that
() is abelian and either or . Hence, in view of Theorem 2.1, we may assume
that and therefore, by Lemma 2.2,
that . We calculate in two ways. Using the defining property of , we obtain ; and using (β), we obtain . Thus, for all . But since , (2.3) holds
in this case as well; therefore for all ,
hence by Lemma 1.1(iii) . Thus, = 0, contrary to our original
hypothesis, so that the case does not in fact occur.

#### Acknowledgment

This research is supported by the Natural Sciences and Engineering Research Council of Canada, Grant no. 3961.

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