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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2008, Article IDΒ 490316, 5 pages
http://dx.doi.org/10.1155/2008/490316
Research Article

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 𝑁 be a 3-prime 2-torsion-free zero-symmetric left near-ring with multiplicative center 𝑍. We prove that if 𝑁 admits a nonzero generalized derivation 𝑓 such that 𝑓(𝑁)βŠ†π‘, then 𝑁 is a commutative ring. We also discuss some related properties.

1. Introduction

Let 𝑁 be a zero-symmetric left near-ring, not necessarily with a multiplicative identity element; and let 𝑍 be its multiplicative center. Define 𝑁 to be 3-prime if for all π‘Ž,π‘βˆˆπ‘β§΅{0}, π‘Žπ‘π‘β‰ {0}; and call 𝑁 2-torsion-free if (𝑁,+) has no elements of order 2. A derivation on 𝑁 is an additive endomorphism 𝐷 of 𝑁 such that 𝐷(π‘₯𝑦)=π‘₯𝐷(𝑦)+𝐷(π‘₯)𝑦 for all π‘₯,π‘¦βˆˆπ‘. A generalized derivation 𝑓 with associated derivation 𝐷 is an additive endomorphism π‘“βˆΆπ‘β†’π‘ such that 𝑓(π‘₯𝑦)=𝑓(π‘₯)𝑦+π‘₯𝐷(𝑦) for all π‘₯,π‘¦βˆˆπ‘. 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 𝐷2=0, then 𝐷=0.

Theorem B. If N is a 3-prime 2-torsion-free near-ring which admits a nonzero derivation D for which 𝐷(𝑁)βŠ†π‘, 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 𝐷(π‘₯)𝐷(𝑦)=𝐷(𝑦)𝐷(π‘₯) for all π‘₯,π‘¦βˆˆπ‘, then N is a commutative ring.

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

We will need three easy lemmas.

Lemma 1.1 (see [1, Lemma 3]). Let N be a 3-prime near-ring.
(i) Ifπ‘§βˆˆπ‘β§΅{0}, then z is not a zero divisor. (ii)If 𝑍⧡{0} contains an element z such that 𝑧+π‘§βˆˆπ‘, then (𝑁,+) is abelian. (iii)If D is a nonzero derivation and π‘₯βˆˆπ‘ is such thatπ‘₯𝐷(𝑁)={0} or 𝐷(𝑁)π‘₯={0}, then π‘₯=0.

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 π‘₯,π‘¦βˆˆπ‘.

Lemma 1.3. Let N be an arbitrary near-ring and let f be a generalized derivation on N with associated derivation D. Then
𝑓(π‘Ž)𝑏+π‘Žπ·(𝑏)𝑐=𝑓(π‘Ž)𝑏𝑐+π‘Žπ·(𝑏)π‘βˆ€π‘Ž,𝑏,π‘βˆˆπ‘.(1.1)

Proof. Clearly 𝑓((π‘Žπ‘)𝑐)=𝑓(π‘Žπ‘)𝑐+π‘Žπ‘π·(𝑐)=(𝑓(π‘Ž)𝑏+π‘Žπ·(𝑏))𝑐+π‘Žπ‘π·(𝑐); and by using Lemma 1.2, we obtain 𝑓(π‘Ž(𝑏𝑐))=𝑓(π‘Ž)𝑏𝑐+π‘Žπ·(𝑏𝑐)=𝑓(π‘Ž)𝑏𝑐+π‘Žπ·(𝑏)𝑐+π‘Žπ‘π·(𝑐).
Comparing these two expressions for 𝑓(π‘Žπ‘π‘) 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 𝑓(𝑁)βŠ†π‘, 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 𝐷≠0. If 𝐷(𝑓(𝑁))={0}, then 𝑓(𝐷(𝑁))={0}.

Proof. We are assuming that 𝐷(𝑓(π‘₯))=0 for all π‘₯βˆˆπ‘. It follows that 𝐷(𝑓(π‘₯𝑦))=𝐷(𝑓(π‘₯)𝑦)+𝐷(π‘₯𝐷(𝑦))=0 for all π‘₯,π‘¦βˆˆπ‘, that is,
𝑓(π‘₯)𝐷(𝑦)+𝐷(π‘₯)𝐷(𝑦)+π‘₯𝐷2(𝑦)=0βˆ€π‘₯,π‘¦βˆˆπ‘.(2.1) Applying 𝐷 again, we get
𝑓(π‘₯)𝐷2(𝑦)+𝐷2(π‘₯)𝐷(𝑦)+𝐷(π‘₯)𝐷2(𝑦)+𝐷(π‘₯)𝐷2(𝑦)+π‘₯𝐷3(𝑦)=0βˆ€π‘₯,π‘¦βˆˆπ‘.(2.2) Taking 𝐷(𝑦) instead of 𝑦 in (2.1) gives 𝑓(π‘₯)𝐷2(𝑦)+𝐷(π‘₯)𝐷2(𝑦)+π‘₯𝐷3(𝑦)=0, hence (2.2) yields
𝐷2(π‘₯)𝐷(𝑦)+𝐷(π‘₯)𝐷2(𝑦)=0βˆ€π‘₯,π‘¦βˆˆπ‘.(2.3) Now, substitute 𝐷(π‘₯) for π‘₯ in (2.1), obtaining 𝑓(𝐷(π‘₯))𝐷(𝑦)+𝐷2(π‘₯)𝐷(𝑦)+𝐷(π‘₯)𝐷2(𝑦)=0; and use (2.3) to conclude that 𝑓(𝐷(π‘₯))𝐷(𝑦)=0 for all π‘₯,π‘¦βˆˆπ‘. Thus, by Lemma 1.1(iii), 𝑓(𝐷(π‘₯))=0 for all π‘₯βˆˆπ‘.

Proof of Theorem 2.1. Since 𝑓≠0, there exists π‘₯βˆˆπ‘ such that0≠𝑓(π‘₯)βˆˆπ‘. Since 𝑓(π‘₯)+𝑓(π‘₯)=𝑓(π‘₯+π‘₯)βˆˆπ‘,(𝑁,+) is abelian by Lemma 1.1(ii). To complete the proof, we show that 𝑁 is multiplicatively commutative.
First, consider the case 𝐷 = 0, so that 𝑓(π‘₯𝑦)=𝑓(π‘₯)π‘¦βˆˆπ‘ for all π‘₯,π‘¦βˆˆπ‘. Then 𝑓(π‘₯)𝑦𝑀=𝑀𝑓(π‘₯)𝑦, hence 𝑓(π‘₯)(π‘¦π‘€βˆ’π‘€π‘¦)=0 for all π‘₯,𝑦,π‘€βˆˆπ‘. Choosing π‘₯ such that 𝑓(π‘₯)β‰ 0 and invoking Lemma 1.1(i), we get 𝑦𝑀 βˆ’ 𝑀𝑦 = 0 for all 𝑦,π‘€βˆˆπ‘.
Now assume that 𝐷≠0, and let π‘βˆˆπ‘β§΅{0}. Then 𝑓(π‘₯𝑐)=𝑓(π‘₯)𝑐+π‘₯𝐷(𝑐)βˆˆπ‘; therefore, (𝑓(π‘₯)𝑐+π‘₯𝐷(𝑐))𝑦=𝑦(𝑓(π‘₯)𝑐+π‘₯𝐷(𝑐)) for all π‘₯,π‘¦βˆˆπ‘, and by Lemma 1.3, we see that 𝑓(π‘₯)𝑐𝑦+π‘₯𝐷(𝑐)𝑦=𝑦𝑓(π‘₯)𝑐+𝑦π‘₯𝐷(𝑐). Since both 𝑓(π‘₯) and 𝐷(𝑐) are in 𝑍, we have 𝐷(𝑐)(π‘₯π‘¦βˆ’π‘¦π‘₯)=0 for all π‘₯,π‘¦βˆˆπ‘, and provided that 𝐷(𝑍)β‰ {0}, we can conclude that 𝑁 is commutative.
Assume now that 𝐷≠0 and 𝐷(𝑍)={0}. In particular, 𝐷(𝑓(π‘₯))=0 for all π‘₯βˆˆπ‘. Note that for π‘βˆˆπ‘ such that 𝑓(𝑐)=0, 𝑓(𝑐π‘₯)=𝑐𝐷(π‘₯)βˆˆπ‘; hence by Lemma 2.2, 𝐷(π‘₯)𝐷(𝑦)βˆˆπ‘ and 𝐷(𝑦)𝐷(π‘₯)βˆˆπ‘ for each π‘₯,π‘¦βˆˆπ‘. If one of these is 0, the other is a central element squaring to 0, hence is also 0. The remaining possibility is that 𝐷(π‘₯)𝐷(𝑦) and 𝐷(𝑦)𝐷(π‘₯) are nonzero central elements, in which case 𝐷(π‘₯) is not a zero divisor. Thus 𝐷(π‘₯)𝐷(π‘₯)𝐷(𝑦)=𝐷(π‘₯)𝐷(𝑦)𝐷(π‘₯) yields 𝐷(π‘₯)(𝐷(π‘₯)𝐷(𝑦)βˆ’π·(𝑦)𝐷(π‘₯))=0=𝐷(π‘₯)𝐷(𝑦)βˆ’π·(𝑦)𝐷(π‘₯). Consequently, 𝑁 is commutative by Theorem C.

3. On Theorems A and C

Theorem C does not extend to generalized derivations, even if 𝑁 is a ring. As in [3], consider the ring 𝐻 of real quaternions, and define π‘“βˆΆπ»β†’π» by 𝑓(π‘₯)=𝑖π‘₯+π‘₯𝑖. It is easy to check that 𝑓 is a generalized derivation with associated derivation given by 𝐷(π‘₯)=π‘₯π‘–βˆ’π‘–π‘₯, and that 𝑓(π‘₯)𝑓(𝑦)=𝑓(𝑦)𝑓(π‘₯) for all π‘₯,π‘¦βˆˆπ».

Theorem A also does not extend to generalized derivations, as we see by letting 𝑁 be the ring 𝑀2(𝐹) of 2 Γ— 2 matrices over a field 𝐹 and letting 𝑓 be defined by 𝑓(π‘₯)=𝑒12π‘₯. 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 𝑓2=0, then𝐷3=0. Moreover, if N is 2-torsion-free, then 𝐷(𝑍)={0}.

Proof. We have 𝑓2(π‘₯𝑦)=𝑓(𝑓(π‘₯)𝑦+π‘₯𝐷(𝑦))=𝑓(π‘₯)𝐷(𝑦)+𝑓(π‘₯)𝐷(𝑦)+π‘₯𝐷2(𝑦)=0βˆ€π‘₯,π‘¦βˆˆπ‘.(3.1) Applying 𝑓 to (3.1) gives
𝑓(π‘₯)𝐷2(𝑦)+𝑓(π‘₯)𝐷2(𝑦)+𝑓(π‘₯)𝐷2(𝑦)+π‘₯𝐷3(𝑦)=0βˆ€π‘₯,π‘¦βˆˆπ‘.(3.2) Substituting 𝐷(𝑦) for 𝑦 in (3.1) gives
𝑓(π‘₯)𝐷2(𝑦)+𝑓(π‘₯)𝐷2(𝑦)+π‘₯𝐷3(𝑦)=0;(3.3) Therefore, by (3.2) and (3.3),
𝑓(π‘₯)𝐷2(𝑦)=0βˆ€π‘₯,π‘¦βˆˆπ‘.(3.4) It now follows from (3.3) that π‘₯𝐷3(𝑦)=0 for all π‘₯,π‘¦βˆˆπ‘; and since 𝑁 is 3-prime, 𝐷3=0.
Suppose now that 𝑁 is 2-torsion-free and that 𝐷(𝑍)β‰ {0}, and let π‘§βˆˆπ‘ be such that 𝐷(𝑧)β‰ 0. Then if π‘₯,π‘¦βˆˆπ‘ and 𝑓(𝑁)π‘₯={0}, then 𝑓(𝑦𝑧)π‘₯=𝑓(𝑦)𝑧π‘₯+𝑦𝐷(𝑧)π‘₯=0=𝑦𝐷(𝑧)π‘₯; and since 𝑁 is 3-prime and 𝐷(𝑧) is not a zero divisor, π‘₯ = 0. It now follows from (3.4) that 𝐷2=0 and hence by Theorem A that 𝐷 = 0. But this contradicts our assumption that 𝐷(𝑍)β‰ {0}, hence 𝐷(𝑍)={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 that 𝑓2=0 and 𝑓(1)βˆˆπ‘, then 𝑓=0.

Proof. Note that 𝑓(π‘₯)=𝑓(1π‘₯)=𝑓(1)π‘₯+1𝐷(π‘₯), so
𝑓(π‘₯)=𝑐π‘₯+𝐷(π‘₯),π‘βˆˆπ‘.(3.5) If 𝑐 = 0, then 𝑓=𝐷 and 𝐷2=0, so 𝐷 = 0 by Theorem A and therefore 𝑓 = 0.
If 𝑐≠0, then 𝑐 is not a zero divisor, hence by (3.4) 𝐷2=0 and 𝐷 = 0. But then 𝑓(π‘₯)=𝑐π‘₯ and 𝑓2(π‘₯)=𝑐2π‘₯=0 for all π‘₯βˆˆπ‘. Since 𝑐2 is not a zero divisor, we get 𝑁={0}β€”a contradiction. Thus, 𝑐 = 0 and we are finished.

4. More on Theorem C

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

𝑓(π‘₯𝑦)=𝐷(π‘₯)𝑦+π‘₯𝑓(𝑦)βˆ€π‘₯,π‘¦βˆˆπ‘.(βˆ—) 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 𝑓 with nonzero associated derivation D such that f satisfies (βˆ—). If 𝑓(π‘₯)𝑓(𝑦)=𝑓(𝑦)𝑓(π‘₯) for all π‘₯,π‘¦βˆˆπ‘, then N is a commutative ring.

Proof. It is correctly shown in [4] that (𝑁,+) is abelian and either 𝑓(𝑁)βŠ†π‘ or 𝐷(𝑓(𝑁))={0}. Hence, in view of Theorem 2.1, we may assume that 𝐷(𝑓(𝑁))=0 and therefore, by Lemma 2.2, that 𝑓(𝐷(𝑁))={0}. We calculate 𝑓(𝐷(π‘₯)𝐷(𝑦)) in two ways. Using the defining property of 𝑓, we obtain 𝑓(𝐷(π‘₯)𝐷(𝑦))=𝑓(𝐷(π‘₯))𝐷(𝑦)+𝐷(π‘₯)𝐷2(𝑦)=𝐷(π‘₯)𝐷2(𝑦); and using (βˆ—), we obtain 𝑓(𝐷(π‘₯)𝐷(𝑦))=𝐷2(π‘₯)𝐷(𝑦)+𝐷(π‘₯)𝑓(𝐷(𝑦))=𝐷2(π‘₯)𝐷(𝑦). Thus, 𝐷2(π‘₯)𝐷(𝑦)=𝐷(π‘₯)𝐷2(𝑦) for all π‘₯,π‘¦βˆˆπ‘. But since 𝐷(𝑓(𝑁))={0}, (2.3) holds in this case as well; therefore 𝐷2(π‘₯)𝐷(𝑦)=0 for all π‘₯,π‘¦βˆˆπ‘, hence by Lemma 1.1(iii) 𝐷2=0. Thus, 𝐷 = 0, contrary to our original hypothesis, so that the case 𝐷(𝑓(𝑁))={0} 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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