Generalized Derivations on Prime Near Rings

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Volume 2013, Article ID 170749,5pages http://dx.doi.org/10.1155/2013/170749

Research Article

Generalized Derivations on Prime Near Rings

Asma Ali,

1

Howard E. Bell,

2

and Phool Miyan

1

1

Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

2

Department of Mathematics, Brock University, St. Catharines, ON, Canada L2S 3A1

Correspondence should be addressed to Asma Ali; asma.ali345@gmail.com

Received 8 October 2012; Accepted 2 January 2013

Academic Editor: Christian Corda

Copyright © 2013 Asma Ali 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.

Let𝑁be a near ring. An additive mapping𝑓 : 𝑁 → 𝑁is said to be a right generalized (resp., left generalized) derivation with associated derivation𝑑on𝑁if𝑓(đ‘„đ‘Š) = 𝑓(đ‘„)𝑩 + đ‘„đ‘‘(𝑩)(resp.,𝑓(đ‘„đ‘Š) = 𝑑(đ‘„)𝑩 + đ‘„đ‘“(𝑩)) for allđ‘„, 𝑩 ∈ 𝑁. A mapping𝑓 : 𝑁 → 𝑁is said to be a generalized derivation with associated derivation𝑑on𝑁if𝑓is both a right generalized and a left generalized derivation with associated derivation𝑑on𝑁. The purpose of the present paper is to prove some theorems in the setting of a semigroup ideal of a 3-prime near ring admitting a generalized derivation, thereby extending some known results on derivations.

1. Introduction

Throughout the paper,𝑁denotes a zero-symmetric left near ring with multiplicative center𝑍, and for any pair of elements

đ‘„, 𝑩 ∈ 𝑁,[đ‘„, 𝑩]denotes the commutatorđ‘„đ‘Š − đ‘Šđ‘„, while the symbol(đ‘„, 𝑩)denotes the additive commutatorđ‘„+đ‘Šâˆ’đ‘„âˆ’đ‘Š. A near ring𝑁is called zero-symmetric if0đ‘„ = 0, for allđ‘„ ∈ 𝑁 (recall that left distributivity yields thatđ‘„0 = 0). The near ring

𝑁is said to be 3-prime ifđ‘„đ‘đ‘Š = {0}forđ‘„, 𝑩 ∈ 𝑁implies that

đ‘„ = 0or𝑩 = 0. A near ring𝑁is called 2-torsion-free if(𝑁, +) has no element of order 2. A nonempty subset𝐮of𝑁is called a semigroup right (resp., semigroup left) ideal if𝐮𝑁 ⊆ 𝐮 (resp.,𝑁𝐮 ⊆ 𝐮), and if𝐮is both a semigroup right ideal and a semigroup left ideal, it is called a semigroup ideal. An additive mapping𝑑 : 𝑁 → 𝑁is a derivation on𝑁if𝑑(đ‘„đ‘Š) = 𝑑(đ‘„)𝑩+

đ‘„đ‘‘(𝑩), for allđ‘„, 𝑩 ∈ 𝑁. An additive mapping𝑓 : 𝑁 → 𝑁

is said to be a right (resp., left) generalized derivation with associated derivation𝑑 if 𝑓(đ‘„đ‘Š) = 𝑓(đ‘„)𝑩 + đ‘„đ‘‘(𝑩) (resp.,

𝑓(đ‘„đ‘Š) = 𝑑(đ‘„)𝑩 + đ‘„đ‘“(𝑩)), for allđ‘„, 𝑩 ∈ 𝑁, and𝑓is said to be a generalized derivation with associated derivation𝑑on𝑁if it is both a right generalized derivation and a left generalized derivation on𝑁with associated derivation𝑑. (Note that this definition differs from the one given by Hvala in [1]; his generalized derivations are our right generalized derivations.) Every derivation on𝑁is a generalized derivation.

In the case of rings, generalized derivations have received significant attention in recent years. We prove some theorems

in the setting of a semigroup ideal of a 3-prime near ring admitting a generalized derivation and thereby extend some known results [2, Theorem 2.1], [3, Theorem 3.1], [4, Theorem 3], and [5, Theorem 3.3].

2. Preliminary Results

We begin with several lemmas, most of which have been proved elsewhere.

Lemma 1 (see [3, Lemma 1.3]). Let𝑁be a 3-prime near ring

and𝑑be a nonzero derivation on𝑁.

(i)If𝑈is a nonzero semigroup right ideal or a nonzero

semigroup left ideal of𝑁, then𝑑(𝑈) Ìž= {0}.

(ii)If𝑈is a nonzero semigroup right ideal of𝑁andđ‘„is

an element of𝑁which centralizes𝑈, thenđ‘„ ∈ 𝑍.

Lemma 2 (see [3, Lemma 1.2]). Let𝑁be a 3-prime near ring.

(i)If𝑧 ∈ 𝑍 \ {0}, then𝑧is not a zero divisor.

(ii)If𝑍 \ {0}contains an element𝑧for which𝑧 + 𝑧 ∈ 𝑍, then(𝑁, +)is abelian.

(2)

Lemma 3 (see [3, Lemmas 1.3 and 1.4]). Let𝑁be a 3-prime

near ring and𝑈be a nonzero semigroup ideal of𝑁. Let𝑑be a

nonzero derivation on𝑁.

(i)Ifđ‘„, 𝑩 ∈ 𝑁andđ‘„đ‘ˆđ‘Š = {0}, thenđ‘„ = 0or𝑩 = 0.

(ii)Ifđ‘„ ∈ 𝑁andđ‘„đ‘ˆ = {0}orđ‘ˆđ‘„ = {0}, thenđ‘„ = 0.

(iii)Ifđ‘„ ∈ 𝑁and𝑑(𝑈)đ‘„ = {0}orđ‘„đ‘‘(𝑈) = {0}, thenđ‘„ = 0.

Lemma 4 (see [3, Lemma 1.5]). If𝑁is a 3-prime near ring

and𝑍contains a nonzero semigroup left ideal or a semigroup

right ideal, then𝑁is a commutative ring.

Lemma 5. If𝑓is a generalized derivation on𝑁with

associ-ated derivation𝑑, then(𝑑(đ‘„)𝑩 + đ‘„đ‘“(𝑩))𝑧 = 𝑑(đ‘„)𝑩𝑧 + đ‘„đ‘“(𝑩)𝑧,

for allđ‘„, 𝑩, 𝑧 ∈ 𝑁.

Proof. We prove only (ii), since (i) is proved in [2]. For all

đ‘„, 𝑩, 𝑧 ∈ 𝑁 we have 𝑓((đ‘„đ‘Š)𝑧) = 𝑓(đ‘„đ‘Š)𝑧 + đ‘„đ‘Šđ‘‘(𝑧) =

(𝑑(đ‘„)𝑩+đ‘„đ‘“(𝑩))𝑧+đ‘„đ‘Šđ‘‘(𝑧)and𝑓(đ‘„(𝑩𝑧)) = 𝑑(đ‘„)𝑩𝑧+đ‘„đ‘“(𝑩𝑧) =

𝑑(đ‘„)𝑩𝑧 + đ‘„đ‘“(𝑓(𝑩)𝑧 + 𝑩𝑑(𝑧)) = 𝑑(đ‘„)𝑩𝑧 + đ‘„đ‘“(𝑩)𝑧 + đ‘„đ‘Šđ‘‘(𝑧). Comparing the two expressions for𝑓(đ‘„đ‘Šđ‘§)gives(ii).

Lemma 6. Let𝑁be a 3-prime near ring and𝑓a generalized

derivation with associated derivation𝑑.

(i)𝑓(đ‘„)𝑩 + đ‘„đ‘‘(𝑩) = đ‘„đ‘‘(𝑩) + 𝑓(đ‘„)𝑩for allđ‘„, 𝑩 ∈ 𝑁.

(ii)𝑑(đ‘„)𝑩 + đ‘„đ‘“(𝑩) = đ‘„đ‘“(𝑩) + 𝑑(đ‘„)𝑩for allđ‘„, 𝑩 ∈ 𝑁.

Proof. (i)𝑓(đ‘„(𝑩 + 𝑩)) = 𝑓(đ‘„)(𝑩 + 𝑩) + đ‘„đ‘‘(𝑩 + 𝑩) = 𝑓(đ‘„)𝑩 +

𝑓(đ‘„)𝑩 + đ‘„đ‘‘(𝑩) + đ‘„đ‘‘(𝑩), and𝑓(đ‘„đ‘Š + đ‘„đ‘Š) = 𝑓(đ‘„)𝑩 + đ‘„đ‘‘(𝑩) +

𝑓(đ‘„)𝑩 + đ‘„đ‘‘(𝑩). Comparing these two equations gives the desired result.

(ii) Again, calculate 𝑓(đ‘„(𝑩 + 𝑩)) and 𝑓(đ‘„đ‘Š + đ‘„đ‘Š) and compare.

Lemma 7. Let𝑁be a 3-prime near ring and𝑓a generalized

derivation with associated derivation𝑑. Then𝑓(𝑍) ⊆ 𝑍.

Proof. Let𝑧 ∈ 𝑍andđ‘„ ∈ 𝑁. Then𝑓(đ‘§đ‘„) = 𝑓(đ‘„đ‘§); that is,

𝑓(𝑧)đ‘„ + 𝑧𝑑(đ‘„) = 𝑑(đ‘„)𝑧 + đ‘„đ‘“(𝑧). ApplyingLemma 6(i), we get

𝑧𝑑(đ‘„) + 𝑓(𝑧)đ‘„ = 𝑑(đ‘„)𝑧 + đ‘„đ‘“(𝑧). It follows that𝑓(𝑧)đ‘„ = đ‘„đ‘“(𝑧) for allđ‘„ ∈ 𝑁, so𝑓(𝑧) ∈ 𝑍.

Lemma 8. Let𝑁be a 3-prime near ring and 𝑈a nonzero

semigroup ideal of𝑁. If𝑓is a nonzero right generalized

deri-vation of𝑁with associated derivation𝑑, then𝑓(𝑈) Ìž= {0}.

Proof. Suppose𝑓(𝑈) = {0}. Then𝑓(đ‘ąđ‘„) = 𝑓(𝑱)đ‘„ + 𝑱𝑑(đ‘„) =

0 = 𝑱𝑑(đ‘„) for all 𝑱 ∈ 𝑈and đ‘„ ∈ 𝑁, and it follows by Lemma 3(ii) that𝑑 = 0. Therefore𝑓(đ‘„đ‘ą) = 𝑓(đ‘„)𝑱 = 0for all𝑱 ∈ 𝑈andđ‘„ ∈ 𝑁, and another appeal toLemma 3(ii) gives𝑓 = 0, which is a contradiction.

Lemma 9 (see [2, Theorem 2.1]). Let𝑁be a 3-prime near ring

with a nonzero right generalized derivation𝑓with associated

derivation𝑑. If𝑓(𝑁) ⊆ 𝑍then(𝑁, +)is abelian. Moreover, if

𝑁is 2-torsion-free, then𝑁is a commutative ring.

Lemma 10 (see [2, Theorem 4.1]). Let𝑁be a 2-torsion-free

3-prime near ring and𝑓a nonzero generalized derivation on

𝑁with associated derivation𝑑. If[𝑓(𝑁), 𝑓(𝑁)] = {0}, then𝑁

is a commutative ring.

3. Main Results

The theorems that we prove in this section extend the results proved in [2, Theorems 2.1 and 3.1], [3, Theorems 2.1, 3.1, and 3.3], and [5, Theorem 3.3].

Theorem 11. Let𝑁be a 3-prime near ring and𝑈be a nonzero

semigroup ideal of 𝑁. Let 𝑓be a nonzero right generalized

derivation with associated derivation𝑑. If𝑓(𝑈) ⊆ 𝑍, then

(𝑁, +)is abelian. Moreover, if𝑁is 2-torsion-free, then𝑁is a commutative ring.

Proof. We begin by showing that(𝑁, +)is abelian, which by

Lemma 2(ii) is accomplished by producing𝑧 ∈ 𝑍 \ {0}such that𝑧 + 𝑧 ∈ 𝑍. Let𝑎be an element of𝑈such that𝑓(𝑎) Ìž= 0. Then for allđ‘„ ∈ 𝑁,đ‘Žđ‘„ ∈ 𝑈andđ‘Žđ‘„ + đ‘Žđ‘„ = 𝑎(đ‘„ + đ‘„) ∈ 𝑈, so that𝑓(đ‘Žđ‘„) ∈ 𝑍and𝑓(đ‘Žđ‘„)+𝑓(đ‘Žđ‘„) ∈ 𝑍; hence we need only to show that there existsđ‘„ ∈ 𝑁such that𝑓(đ‘Žđ‘„) Ìž= 0. Suppose that this is not the case, so that𝑓((đ‘Žđ‘„)𝑎) = 0 = 𝑓(đ‘Žđ‘„)𝑎+đ‘Žđ‘„đ‘‘(𝑎) =

đ‘Žđ‘„đ‘‘(𝑎), for allđ‘„ ∈ 𝑁. ByLemma 3(i) either𝑎 = 0or𝑑(𝑎) = 0. If𝑑(𝑎) = 0, then𝑓(đ‘„đ‘Ž) = 𝑓(đ‘„)𝑎 + đ‘„đ‘‘(𝑎); that is, 𝑓(đ‘„đ‘Ž) =

𝑓(đ‘„)𝑎 ∈ 𝑍, for allđ‘„ ∈ 𝑁. Thus[𝑓(𝑱)𝑎, 𝑩] = 0, for all𝑩 ∈ 𝑁, and𝑱 ∈ 𝑈. This implies that𝑓(𝑱)[𝑎, 𝑩] = 0, for all𝑱 ∈ 𝑈 and𝑩 ∈ 𝑁andLemma 2(i) gives𝑎 ∈ 𝑍. Thus0 = 𝑓(đ‘Žđ‘„) =

𝑓(đ‘„đ‘Ž) = 𝑓(đ‘„)𝑎, for allđ‘„ ∈ 𝑁. Replacingđ‘„by𝑱 ∈ 𝑈, we have

𝑓(𝑈)𝑎 = 0, and by Lemmas2(i) and8, we get that𝑎 = 0. Thus, we have a contradiction.

To complete the proof, we show that if𝑁is 2-torsion-free, then𝑁is commutative.

Consider first the case𝑑 = 0. This implies that𝑓(đ‘ąđ‘„) =

𝑓(𝑱)đ‘„ ∈ 𝑍for all𝑱 ∈ 𝑈andđ‘„ ∈ 𝑁. ByLemma 8, we have

𝑱 ∈ 𝑈such that𝑓(𝑱) ∈ 𝑍 \ {0}, so𝑁is commutative by Lemma 2(iii).

Now consider the case𝑑 Ìž= 0. Let𝑐 ∈ 𝑍 \ {0}. This implies that ifđ‘„ ∈ 𝑈,𝑓(đ‘„đ‘) = 𝑓(đ‘„)𝑐 + đ‘„đ‘‘(𝑐) ∈ 𝑍. Thus(𝑓(đ‘„)𝑐 + đ‘„đ‘‘(𝑐))𝑩 = 𝑩(𝑓(đ‘„)𝑐 + đ‘„đ‘‘(𝑐))for allđ‘„, 𝑩 ∈ 𝑈and 𝑐 ∈ 𝑍. Therefore byLemma 5(i),𝑓(đ‘„)𝑐𝑩+đ‘„đ‘‘(𝑐)𝑩 = 𝑩𝑓(đ‘„)𝑐+đ‘Šđ‘„đ‘‘(𝑐)

for allđ‘„, 𝑩 ∈ 𝑈and𝑐 ∈ 𝑍. Since𝑑(𝑐) ∈ 𝑍and𝑓(đ‘„) ∈ 𝑍, we obtain𝑑(𝑐)[đ‘„, 𝑩] = 0, for allđ‘„, 𝑩 ∈ 𝑈and𝑐 ∈ 𝑍. Let𝑑(𝑍) Ìž= 0. Choosing𝑐such that𝑑(𝑐) Ìž= 0and noting that𝑑(𝑐)is not a zero divisor, we have[đ‘„, 𝑩] = 0for allđ‘„, 𝑩 ∈ 𝑈. ByLemma 1(ii),

𝑈 ⊆ 𝑍; hence𝑁is commutative byLemma 4.

The remaining case is𝑑 Ìž= 0and𝑑(𝑍) = {0}. Suppose we can show that𝑈 ∩ 𝑍 Ìž= {0}. Taking𝑧 ∈ (𝑈 ∩ 𝑍) \ {0}andđ‘„ ∈

𝑁, we have𝑓(đ‘„đ‘§) = 𝑓(đ‘„)𝑧 ∈ 𝑍; therefore𝑓(𝑁) ⊆ 𝑍by Lemma 2(iii) and𝑁is commutative byLemma 9.

Assume, then, that𝑈 ∩ 𝑍 = {0}. For each𝑱 ∈ 𝑈,𝑓(𝑱2) = 𝑓(𝑱)𝑱 + 𝑱𝑑(𝑱) = 𝑱(𝑓(𝑱) + 𝑑(𝑱)) ∈ 𝑈 ∩ 𝑍, so𝑓(𝑱2) = 0. Thus, for all𝑱 ∈ 𝑈andđ‘„ ∈ 𝑁,𝑓(𝑱2đ‘„) = 𝑓(𝑱2)đ‘„ + 𝑱2𝑑(đ‘„) =

𝑱2𝑑(đ‘„) ∈ 𝑈 ∩ 𝑍, so𝑱2𝑑(đ‘„) = 0, and byLemma 3(iii)𝑱2= 0.

Since 𝑓(đ‘„đ‘ą) = 𝑓(đ‘„)𝑱 + đ‘„đ‘‘(𝑱) ∈ 𝑍 for all𝑱 ∈ 𝑈 and

đ‘„ ∈ 𝑁, we have (𝑓(đ‘„)𝑱 + đ‘„đ‘‘(𝑱))𝑱 = 𝑱(𝑓(đ‘„)𝑱 + đ‘„đ‘‘(𝑱)), and right multiplying by𝑱givesđ‘ąđ‘„đ‘‘(𝑱)𝑱 = 0. Consequently

𝑑(𝑱)𝑱𝑁𝑑(𝑱)𝑱 = {0}, so that𝑑(𝑱)𝑱 = 0for all𝑱 ∈ 𝑈. Since

(3)

𝑓(𝑱)𝑱 = 0for all𝑱 ∈ 𝑈. But byLemma 8, there exists𝑱0∈ 𝑈 for which𝑓(𝑱0) Ìž= 0; and since𝑓(𝑱0) ∈ 𝑍, we get𝑱0 = 0—a contradiction. Therefore𝑈 ∩ 𝑍 Ìž= {0}as required.

Theorem 12. Let 𝑁be a 3-prime near ring with a nonzero

generalized derivation𝑓with associated nonzero derivation𝑑.

Let𝑈be a nonzero semigroup ideal of𝑁. If[𝑓(𝑈), 𝑓(𝑈)] = 0,

then(𝑁, +)is abelian.

Proof. Assume thatđ‘„ ∈ 𝑁is such that [đ‘„, 𝑓(𝑈)] = [đ‘„ +

đ‘„, 𝑓(𝑈)] = 0. For all 𝑱, 𝑣 ∈ 𝑈 such that 𝑱 + 𝑣 ∈ 𝑈,

[đ‘„ + đ‘„, 𝑓(𝑱 + 𝑣)] = 0. This implies that

(đ‘„ + đ‘„) 𝑓 (𝑱 + 𝑣) = 𝑓 (𝑱 + 𝑣) (đ‘„ + đ‘„) , (đ‘„ + đ‘„) 𝑓 (𝑱) + (đ‘„ + đ‘„) 𝑓 (𝑣)

= 𝑓 (𝑱 + 𝑣) đ‘„ + 𝑓 (𝑱 + 𝑣) đ‘„, 𝑓 (𝑱) (đ‘„ + đ‘„) + 𝑓 (𝑣) (đ‘„ + đ‘„)

= đ‘„đ‘“ (𝑱 + 𝑣) + đ‘„đ‘“ (𝑱 + 𝑣) , 𝑓 (𝑱) đ‘„ + 𝑓 (𝑱) đ‘„ + 𝑓 (𝑣) đ‘„ + 𝑓 (𝑣) đ‘„

= đ‘„đ‘“ (𝑱) + đ‘„đ‘“ (𝑣) + đ‘„đ‘“ (𝑱) + đ‘„đ‘“ (𝑣) , đ‘„đ‘“ (𝑱) + đ‘„đ‘“ (𝑱) + đ‘„đ‘“ (𝑣) + đ‘„đ‘“ (𝑣)

= đ‘„đ‘“ (𝑱) + đ‘„đ‘“ (𝑣) + đ‘„đ‘“ (𝑱) + đ‘„đ‘“ (𝑣) , đ‘„ (𝑓 (𝑱) + 𝑓 (𝑣) − 𝑓 (𝑱) − 𝑓 (𝑣)) = 0,

đ‘„đ‘“ (𝑱 + 𝑣 − 𝑱 − 𝑣) = 0.

(1)

This equation may be restated asđ‘„đ‘“(𝑐) = 0, where𝑐 = (𝑱, 𝑣). Let𝑎, 𝑏 ∈ 𝑈. Then𝑎𝑏 ∈ 𝑈and𝑎𝑏 + 𝑎𝑏 = 𝑎(𝑏 + 𝑏) ∈ 𝑈, so[𝑓(𝑎𝑏), 𝑓(𝑈)] = {0} = [𝑓(𝑎𝑏) + 𝑓(𝑎𝑏), 𝑓(𝑈)], and by the argument in the previous paragraph,𝑓(𝑎𝑏)𝑓(𝑐) = 0. We now have𝑓(𝑈2)𝑓(𝑱, 𝑣) = 0for all𝑱, 𝑣 ∈ 𝑈such that𝑱 + 𝑣 ∈ 𝑈. Taking đ‘€ ∈ 𝑈2 and đ‘„ ∈ 𝑁, we get 𝑓(đ‘„đ‘€)𝑓((𝑱, 𝑣)) =

(𝑑(đ‘„)đ‘€ + đ‘„đ‘“(đ‘€))𝑓((𝑱, 𝑣)) = 0 = 𝑑(đ‘„)đ‘€đ‘“((𝑱, 𝑣)), and since

𝑈2is a nonzero semigroup ideal by Lemma 3(ii) and𝑑 Ìž= 0,

Lemma 3(i) gives

𝑓 ((𝑱, 𝑣)) = 0 ∀𝑱, 𝑣 ∈ 𝑈such that𝑱 + 𝑣 ∈ 𝑈. (2)

Take𝑱 = 𝑟𝑩and𝑣 = 𝑟𝑧, where𝑟 ∈ 𝑈and𝑩, 𝑧 ∈ 𝑁, so that

𝑱 + 𝑣 = 𝑟𝑩 + 𝑟𝑧 = 𝑟(𝑩 + 𝑧) ∈ 𝑈. By (2) we have

𝑓 ((𝑟𝑩, 𝑟𝑧)) = 𝑓 (𝑟 (𝑩, 𝑧)) = 0 ∀ 𝑟 ∈ 𝑈, 𝑩, 𝑧 ∈ 𝑁. (3)

Replacing𝑟byđ‘Ÿđ‘€,đ‘€ ∈ 𝑈, we obtain𝑓(𝑟(đ‘€đ‘Š, đ‘€đ‘§)) = 0 =

𝑑(𝑟)(đ‘€(𝑩, 𝑧)) + 𝑟𝑓((đ‘€đ‘Š, đ‘€đ‘§)); so by (3)𝑑(𝑟)𝑈(𝑩, 𝑧) = 0for all

𝑟 ∈ 𝑈and𝑩, 𝑧 ∈ 𝑁. It follows immediately by Lemmas1(i) and3(i) that(𝑩, 𝑧) = 0for all𝑩, 𝑧 ∈ 𝑁; that is,(𝑁, +)is abelian.

Theorem 13. Let 𝑁 be a 2-torsion-free 3-prime near ring

and𝑈be a nonzero semigroup ideal of𝑁. If𝑓is a nonzero

generalized derivation with associated derivation𝑑such that

[𝑓(𝑈), 𝑓(𝑈)] = 0, then𝑁is a commutative ring if it satisfies

one of the following: (i)𝑑(𝑍) Ìž= {0}; (ii)𝑈 ∩ 𝑍 Ìž= {0}; (iii)𝑑 = 0

and𝑓(𝑍) Ìž= {0}.

Proof. (i) Let𝑎 ∈ 𝑁centralizes 𝑓(𝑈), and let𝑧 ∈ 𝑍 such

that𝑑(𝑧) Ìž= 0. Then a centralizes𝑓(𝑱𝑧)for all𝑱 ∈ 𝑈, so that

𝑎(𝑓(𝑱)𝑧 + 𝑱𝑑(𝑧)) = (𝑓(𝑱)𝑧 + 𝑱𝑑(𝑧))𝑎and𝑎𝑱𝑑(𝑧) = 𝑱𝑑(𝑧)𝑎. Since𝑑(𝑧) ∈ 𝑍 \ {0},𝑑(𝑧)[𝑎, 𝑱] = 0 = [𝑎, 𝑱]for all𝑱 ∈ 𝑈. Therefore a centralizes𝑈, and byLemma 1(ii),𝑎 ∈ 𝑍. Since

𝑓(𝑈)centralizes𝑓(𝑈),𝑓(𝑈) ⊆ 𝑍and our result follows by Theorem 11.

(ii) We may assume𝑑(𝑍) = {0}. Let𝑧 ∈ (𝑈 ∩ 𝑍) \ {0}. Then for allđ‘„, 𝑩 ∈ 𝑁,𝑓(đ‘„đ‘§) = 𝑓(đ‘„)𝑧and𝑓(𝑩𝑧) = 𝑓(𝑩)𝑧 commute; hence 𝑧2[𝑓(đ‘„), 𝑓(𝑩)] = 0 = [𝑓(đ‘„), 𝑓(𝑩)]. Our result now follows fromLemma 10.

(iii) Let𝑱, 𝑣 ∈ 𝑈and 𝑧 ∈ 𝑍such that𝑓(𝑧) Ìž= 0. Then

[𝑓(𝑧𝑱), 𝑓(𝑱)] = 0 = [𝑓(𝑧)𝑱, 𝑓(𝑣)], and since𝑓(𝑧) ∈ 𝑍 \ {0},

𝑓(𝑧)[𝑱, 𝑓(𝑱)] = 0 = [𝑱, 𝑓(𝑣)]. Thus 𝑓(𝑈) centralizes 𝑈, and byLemma 1(ii),𝑓(𝑈) ⊆ 𝑍. Our result now follows by Theorem 11.

We have already observed that if 𝑓 is a generalized derivation with𝑑 = 0, then𝑓(đ‘„)𝑩 = đ‘„đ‘“(𝑩)for allđ‘„, 𝑩 ∈ 𝑁. For 3-prime near rings, we have the following converse.

Theorem 14. Let𝑁be a 3-prime near ring and𝑈be a nonzero

semigroup ideal of 𝑁. If 𝑓 is a nonzero right generalized

derivation of 𝑁with associated derivation 𝑑 and𝑓(đ‘„)𝑩 =

đ‘„đ‘“(𝑩), for allđ‘„, 𝑩 ∈ 𝑈, then𝑑 = 0.

Proof. We are given that𝑓(đ‘„)𝑩 = đ‘„đ‘“(𝑩)for allđ‘„, 𝑩 ∈ 𝑈.

Substituting𝑩𝑧for𝑩, we get𝑓(đ‘„)𝑩𝑧 = đ‘„đ‘“(𝑩𝑧) = đ‘„(𝑓(𝑩)𝑧 + 𝑩𝑑(𝑧))for all đ‘„, 𝑩, 𝑧 ∈ 𝑈. It follows thatđ‘„đ‘Šđ‘‘(𝑧) = 0 for allđ‘„, 𝑩, 𝑧 ∈ 𝑈; that is,đ‘„đ‘ˆđ‘‘(𝑧) = {0}for allđ‘„, 𝑧 ∈ 𝑈. By Lemma 3(i),𝑑(𝑈) = 0, and hence𝑑 = 0byLemma 1(i).

4. Generalized Derivations

Acting as a Homomorphism or

an Antihomomorphism

In [4], Bell and Kappe proved that if𝑅is a semiprime ring and

𝑑is a derivation on𝑅which is either an endomorphism or an antiendomorphism on𝑅, then𝑑 = 0. Of course, derivations which are not endomorphisms or antiendomorphisms on𝑅 may behave as such on certain subsets of𝑅; for example, any derivation𝑑behaves as the zero endomorphism on the subring đ¶ consisting of all constants (i.e., the elements đ‘„ for which 𝑑(đ‘„) = 0). In fact in a semiprime ring 𝑅, 𝑑 may behave as an endomorphism on a proper ideal of𝑅. However as noted in [4], the behaviour of𝑑 is somewhat restricted in the case of a prime ring. Recently the authors in [6] considered(𝜃, 𝜙)-derivation𝑑acting as a homomorphism or an antihomomorphism on a nonzero Lie ideal of a prime ring and concluded that𝑑 = 0. In this section we establish similar results in the setting of a semigroup ideal of a 3-prime near ring admitting a generalized derivation.

Theorem 15. Let𝑁be a 3-prime near ring and𝑈be a

non-zero semigroup ideal of 𝑁. Let 𝑓be a nonzero generalized

(4)

homomorphism on𝑈, then 𝑓is the identity map on𝑁and

𝑑 = 0.

Proof. By the hypothesis

𝑓 (đ‘„đ‘Š) = 𝑑 (đ‘„) 𝑩 + đ‘„đ‘“ (𝑩) = 𝑓 (đ‘„) 𝑓 (𝑩) âˆ€đ‘„, 𝑩 ∈ 𝑈.

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Replacing𝑩by𝑩𝑧in the above relation, we get

𝑓 (đ‘„đ‘Šđ‘§) = 𝑑 (đ‘„) 𝑩𝑧 + đ‘„đ‘“ (𝑩𝑧) âˆ€đ‘„, 𝑩, 𝑧 ∈ 𝑈, (5)

or

𝑓 (đ‘„đ‘Š) 𝑓 (𝑧) = 𝑑 (đ‘„) 𝑩𝑧 + đ‘„ (𝑑 (𝑩) 𝑧 + 𝑩𝑓 (𝑧)) âˆ€đ‘„, 𝑩, 𝑧 ∈ 𝑈. (6)

This implies that

(𝑑 (đ‘„) 𝑩 + đ‘„đ‘“ (𝑩)) 𝑓 (𝑧) = 𝑑 (đ‘„) 𝑩𝑧 + đ‘„đ‘‘ (𝑩) 𝑧 + đ‘„đ‘Šđ‘“ (𝑧) âˆ€đ‘„, 𝑩, 𝑧 ∈ 𝑈.

(7)

UsingLemma 5(ii), we get

𝑑 (đ‘„) 𝑩𝑓 (𝑧) + đ‘„đ‘“ (𝑩) 𝑓 (𝑧)

= 𝑑 (đ‘„) 𝑩𝑧 + đ‘„đ‘‘ (𝑩) 𝑧 + đ‘„đ‘Šđ‘“ (𝑧) âˆ€đ‘„, 𝑩, 𝑧 ∈ 𝑈, (8)

or

𝑑 (đ‘„) 𝑩𝑓 (𝑧) + đ‘„đ‘“ (𝑩𝑧)

= 𝑑 (đ‘„) 𝑩𝑧 + đ‘„đ‘‘ (𝑩) 𝑧 + đ‘„đ‘Šđ‘“ (𝑧) âˆ€đ‘„, 𝑩, 𝑧 ∈ 𝑈. (9)

This implies that

𝑑 (đ‘„) 𝑩𝑓 (𝑧) + đ‘„đ‘‘ (𝑩) 𝑧 + đ‘„đ‘Šđ‘“ (𝑧)

= 𝑑 (đ‘„) 𝑩𝑧 + đ‘„đ‘‘ (𝑩) 𝑧 + đ‘„đ‘Šđ‘“ (𝑧) âˆ€đ‘„, 𝑩, 𝑧 ∈ 𝑈. (10)

That is,

𝑑 (đ‘„) 𝑩𝑓 (𝑧) = 𝑑 (đ‘„) 𝑩𝑧 âˆ€đ‘„, 𝑩, 𝑧 ∈ 𝑈. (11)

Therefore

𝑑 (đ‘„) 𝑩 (𝑓 (𝑧) − 𝑧) = 0 âˆ€đ‘„, 𝑩, 𝑧 ∈ 𝑈, (12)

which implies that

𝑑 (đ‘„) 𝑈 (𝑓 (𝑧) − 𝑧) = {0} âˆ€đ‘„, 𝑧 ∈ 𝑈. (13)

It follows byLemma 3(i) that either𝑑(𝑈) = 0or𝑓(𝑧) = 𝑧for all𝑧 ∈ 𝑈.

In fact, as we now show, both of these conditions hold. Suppose that𝑓(𝑱) = 𝑱for all𝑱 ∈ 𝑈. Then for all𝑱 ∈ 𝑈 andđ‘„ ∈ 𝑁,𝑓(đ‘„đ‘ą) = đ‘„đ‘ą = 𝑑(đ‘„)𝑱 + đ‘„đ‘“(𝑱) = 𝑑(đ‘„)𝑱 + đ‘„đ‘ą; hence𝑑(đ‘„)𝑈 = {0}for allđ‘„ ∈ 𝑁, and thus𝑑 = 0.

On the other hand, suppose that𝑑(𝑈) = {0}, so that𝑑 = 0. Then for allđ‘„, 𝑩 ∈ 𝑈,𝑓(đ‘„đ‘Š) = 𝑓(đ‘„)𝑩 = 𝑓(đ‘„)𝑓(𝑩), so that

𝑓(đ‘„)(𝑩 − 𝑓(𝑩)) = 0. Replacing𝑩by𝑧𝑩,𝑧 ∈ 𝑁, and noting

that𝑓(𝑧𝑩) = 𝑧𝑓(𝑩), we see that𝑓(đ‘„)𝑁(𝑩 − 𝑓(𝑩)) = {0}for all

đ‘„, 𝑩 ∈ 𝑈. Therefore,𝑓(𝑈) = {0}or𝑓is the identity on𝑈. But

𝑓(𝑈) = {0}contradictsLemma 8, so𝑓is the identity on𝑈. We now know that𝑓is the identity on𝑈and𝑓(đ‘„đ‘Š) =

đ‘„đ‘“(𝑩)for allđ‘„, 𝑩 ∈ 𝑁. Consequently,𝑓(đ‘ąđ‘„) = đ‘ąđ‘„ = 𝑱𝑓(đ‘„)

for all𝑱 ∈ 𝑈andđ‘„ ∈ 𝑁, so that𝑈(đ‘„ − 𝑓(đ‘„)) = {0}for all

đ‘„ ∈ 𝑁. It follows that𝑓is the identity on𝑁.

Theorem 16. Let 𝑁 be a 3-prime near ring and 𝑈 be a

nonzero semigroup ideal of𝑁. If𝑓is a nonzero generalized

derivation on𝑁with associated derivation𝑑. If𝑓acts as an

antihomomorphism on𝑈, then𝑑 = 0,𝑓is the identity map on

𝑁, and𝑁is a commutative ring.

Proof. We begin by showing that𝑑 = 0if and only if𝑓is the

identity map on𝑁.

Clearly if𝑓is the identity map on𝑁,đ‘„đ‘‘(𝑩) = 0for all

đ‘„, 𝑩 ∈ 𝑁, and hence𝑑 = 0.

Conversely, assume that𝑑 = 0, in which case𝑓(đ‘„đ‘Š) =

𝑓(đ‘„)𝑩 = đ‘„đ‘“(𝑩)for allđ‘„, 𝑩 ∈ 𝑁. It follows that for anyđ‘„, 𝑩, 𝑧 ∈

𝑈,

𝑓 (đ‘Šđ‘„đ‘§) = 𝑓 (𝑧) 𝑓 (đ‘Šđ‘„) = 𝑓 (𝑧) 𝑩𝑓 (đ‘„)

= 𝑓 (𝑧𝑩) 𝑓 (đ‘„) = 𝑧𝑓 (𝑩) 𝑓 (đ‘„) = 𝑧𝑓 (đ‘„đ‘Š) . (14)

On the other hand,

𝑓 (đ‘Šđ‘„đ‘§) = 𝑓 (đ‘„đ‘§) 𝑓 (𝑩) = 𝑓 (đ‘„) 𝑧𝑓 (𝑩) = 𝑓 (đ‘„) 𝑓 (𝑧𝑩) = 𝑓 (đ‘„) 𝑓 (𝑩) 𝑓 (𝑧) = 𝑓 (đ‘Šđ‘„) 𝑓 (𝑧) = 𝑓 (𝑩) đ‘„đ‘“ (𝑧) = 𝑓 (𝑩) 𝑓 (đ‘„đ‘§) = 𝑓 (𝑩) 𝑓 (đ‘„) 𝑧 = 𝑓 (đ‘„đ‘Š) 𝑧.

(15)

Comparing (14) and (15) shows that𝑓(𝑈2)centralizes𝑈, so that𝑓(𝑈2) ⊆ 𝑍byLemma 1(ii).

Now 𝑈2 is a nonzero semigroup ideal by Lemma 3(ii); hence𝑓(𝑈2) Ìž= 0byLemma 8. Choosingđ‘„, 𝑩 ∈ 𝑈such that

𝑓(đ‘„đ‘Š) Ìž= 0, we see that for any𝑧 ∈ 𝑈,𝑓(đ‘„đ‘Š)𝑧 = 𝑓(đ‘„đ‘Šđ‘§) =

𝑓(𝑩𝑧)𝑓(đ‘„) = 𝑓(𝑩)𝑧𝑓(đ‘„) = 𝑓(𝑩)𝑓(đ‘§đ‘„) = 𝑓(𝑩)𝑓(đ‘„)𝑓(𝑧) = 𝑓(đ‘„đ‘Š)𝑓(𝑧), and hence𝑓(đ‘„đ‘Š)(𝑧 − 𝑓(𝑧)) = 0. Since𝑓(đ‘„đ‘Š) ∈

𝑍 \ {0}, we conclude that𝑓(𝑧) = 𝑧for all𝑧 ∈ 𝑈, and it follows easily that𝑓is the identity map on𝑁.

We note now that if the identity map on 𝑁acts as an antihomomorphism on𝑈, then𝑈is commutative, so that by Lemmas1(ii) and4𝑁is a commutative ring.

To complete the proof of our theorem, we need only to argue that𝑑 = 0. By our antihomomorphism hypothesis

𝑓 (đ‘„đ‘Š) = 𝑑 (đ‘„) 𝑩 + đ‘„đ‘“ (𝑩) = 𝑓 (𝑩) 𝑓 (đ‘„) âˆ€đ‘„, 𝑩 ∈ 𝑈. (16)

Replacing𝑩byđ‘„đ‘Šin the above relation, we get

𝑓 (đ‘„đ‘Š) 𝑓 (đ‘„) = 𝑓 (đ‘„đ‘„đ‘Š) = 𝑑 (đ‘„) đ‘„đ‘Š + đ‘„đ‘“ (đ‘„đ‘Š) âˆ€đ‘„, 𝑩 ∈ 𝑈.

(17)

This implies that

(𝑑 (đ‘„) 𝑩 + đ‘„đ‘“ (𝑩)) 𝑓 (đ‘„)

(5)

UsingLemma 5(ii), we get

𝑑 (đ‘„) 𝑩𝑓 (đ‘„) + đ‘„đ‘“ (𝑩) 𝑓 (đ‘„)

= 𝑑 (đ‘„) đ‘„đ‘Š + đ‘„đ‘“ (𝑩) 𝑓 (đ‘„) âˆ€đ‘„, 𝑩 ∈ 𝑈. (19)

Thus

𝑑 (đ‘„) 𝑩𝑓 (đ‘„) = 𝑑 (đ‘„) đ‘„đ‘Š âˆ€đ‘„, 𝑩 ∈ 𝑈. (20)

Replacing𝑩by𝑩𝑟in (20) and using (20), we get

𝑑 (đ‘„) 𝑩𝑟𝑓 (đ‘„) = 𝑑 (đ‘„) đ‘„đ‘Šđ‘Ÿ, and so𝑑 (đ‘„) 𝑩 [𝑟, 𝑓 (đ‘„)] = 0

âˆ€đ‘„, 𝑩 ∈ 𝑈, 𝑟 ∈ 𝑁.

(21)

Application ofLemma 3(i) yields that for eachđ‘„ ∈ 𝑈either

𝑑(đ‘„) = 0or[𝑟, 𝑓(đ‘„)] = 0; that is𝑑(đ‘„) = 0or𝑓(đ‘„) ∈ 𝑍. Suppose that there existsđ‘€ ∈ 𝑈such that𝑓(đ‘€) ∈ 𝑍 \ {0}. Then for all𝑣 ∈ 𝑈such that𝑑(𝑣) = 0,𝑓(đ‘€đ‘Ł) = 𝑓(đ‘€)𝑣 =

𝑓(𝑣)𝑓(đ‘€) = 𝑓(đ‘€)𝑓(𝑣), and hence𝑓(đ‘€)(𝑣 − 𝑓(𝑣)) = 0 =

𝑣−𝑓(𝑣). Now consider arbitraryđ‘„, 𝑩 ∈ 𝑈. If one of𝑓(đ‘„), 𝑓(𝑩) is in𝑍, then𝑓(đ‘„đ‘Š) = 𝑓(đ‘„)𝑓(𝑩). If𝑑(đ‘„) = 0 = 𝑑(𝑩), then

𝑑(đ‘„đ‘Š) = 𝑑(đ‘„)𝑩 + đ‘„đ‘‘(𝑩) = 0, so𝑓(đ‘„đ‘Š) = đ‘„đ‘Š = 𝑓(đ‘„)𝑓(𝑩). Therefore 𝑓(đ‘„đ‘Š) = 𝑓(đ‘„)𝑓(𝑩) for all đ‘„, 𝑩 ∈ 𝑈, and by Theorem 15,𝑓is the identity map on𝑁, and therefore𝑑 = 0. The remaining possibility is that for eachđ‘„ ∈ 𝑈, either

𝑑(đ‘„) = 0or𝑓(đ‘„) = 0. Let𝑱 ∈ 𝑈 \ {0}, and let𝑈1= 𝑱𝑁. Then

𝑈1is a nonzero semigroup right ideal contained in𝑈and𝑈1 is an additive subgroup of𝑁. The sets{đ‘„ ∈ 𝑈1|𝑑(đ‘„) = 0}and

{đ‘„ ∈ 𝑈1|𝑓(đ‘„) = 0}are additive subgroups of𝑈1with union equal to𝑈1, so𝑑(𝑈1) = {0}or𝑓(𝑈1) = {0}. If𝑑(𝑈1) = {0}, then𝑑 = 0byLemma 1(i). Suppose, then, that𝑓(𝑈1) = {0}. Then for arbitraryđ‘„, 𝑩 ∈ 𝑁 𝑓(đ‘ąđ‘„đ‘Š) = 𝑓(đ‘ąđ‘„)𝑩+đ‘ąđ‘„đ‘‘(𝑩) = 0 =

đ‘ąđ‘„đ‘‘(𝑩), so𝑱𝑁𝑑(𝑩) = {0}, and again𝑑 = 0. This completes the proof.

Acknowledgments

A. Ali and P. Miyan gratefully acknowledge the support received by the University Grants Commission, India Grant F. no. 33–106 (2008) and Council of Scientific and Industrial Research, India Grant F. no. 9/112 (0475) 2 K12-EMR-I. Let

𝑧 ∈ 𝑍andđ‘„ ∈ 𝑁. Then𝑓(đ‘§đ‘„) = 𝑓(đ‘„đ‘§); that is,𝑓(𝑧)đ‘„ + 𝑧𝑑(đ‘„) = 𝑑(đ‘„)𝑧 + đ‘„đ‘“(𝑧).

References

[1] B. Hvala, “Generalized derivations in rings,”Communications in

Algebra, vol. 26, no. 4, pp. 1147–1166, 1998.

[2] H. E. Bell, “On prime near-rings with generalized derivation,”

International Journal of Mathematics and Mathematical

Sci-ences, vol. 2008, Article ID 490316, 5 pages, 2008.

[3] H. E. Bell, “On derivations in near rings II,” inNearrings,

Nearfields and K-Loops, vol. 426 of Mathematical

Applica-tions, pp. 191–197, Kluwer Academic Publishers, Dordrecht, The

Netherlands, 1997.

[4] H. E. Bell and L. C. Kappe, “Rings in which derivations satisfy certain algebraic conditions,”Acta Mathematica Hungarica, vol. 53, no. 3-4, pp. 339–346, 1989.

[5] H. E. Bell and G. Mason, “On derivations in near-rings,” in

Near-Rings and Near-Fields, vol. 137 of North-Holland

Math-ematical Studies, pp. 31–35, North-Holland, Amsterdam, The

Netherlands, 1987.

[6] A. Asma, N. Rehman, and A. Shakir, “On Lie ideals with derivations as homomorphisms and antihomomorphisms,”

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