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International Journal of Mathematics and Mathematical Sciences
Volume 2013 (2013), Article ID 170749, 5 pages
http://dx.doi.org/10.1155/2013/170749
Research Article

Generalized Derivations on Prime Near Rings

1Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2Department of Mathematics, Brock University, St. Catharines, ON, Canada L2S 3A1

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.

Abstract

Let be a near ring. An additive mapping is said to be a right generalized (resp., left generalized) derivation with associated derivation on if (resp., ) for all . A mapping is said to be a generalized derivation with associated derivation on if is both a right generalized and a left generalized derivation with associated derivation on . 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, denotes a zero-symmetric left near ring with multiplicative center , and for any pair of elements , denotes the commutator , while the symbol denotes the additive commutator . A near ring is called zero-symmetric if , for all (recall that left distributivity yields that ). The near ring is said to be 3-prime if for implies that or . A near ring is called 2-torsion-free if has no element of order 2. A nonempty subset of is called a semigroup right (resp., semigroup left) ideal if (resp., ), and if is both a semigroup right ideal and a semigroup left ideal, it is called a semigroup ideal. An additive mapping is a derivation on if , for all . An additive mapping is said to be a right (resp., left) generalized derivation with associated derivation if (resp., ), for all , and is said to be a generalized derivation with associated derivation on if it is both a right generalized derivation and a left generalized derivation on with associated derivation . (Note that this definition differs from the one given by Hvala in [1]; his generalized derivations are our right generalized derivations.) Every derivation on is 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]). Let be a 3-prime near ring and be a nonzero derivation on .(i)If is a nonzero semigroup right ideal or a nonzero semigroup left ideal of , then .(ii)If is a nonzero semigroup right ideal of and is an element of which centralizes , then .

Lemma 2 (see [3, Lemma 1.2]). Let be a 3-prime near ring.(i) If, then is not a zero divisor.(ii) If contains an element for which , then is abelian.(iii) If and is an element of such that , then .

Lemma 3 (see [3, Lemmas 1.3 and 1.4]). Let be a 3-prime near ring and be a nonzero semigroup ideal of . Let be a nonzero derivation on .(i) If and , then or .(ii) If and or , then .(iii) If and or , then .

Lemma 4 (see [3, Lemma 1.5]). If is a 3-prime near ring and contains a nonzero semigroup left ideal or a semigroup right ideal, then is a commutative ring.

Lemma 5. If is a generalized derivation on with associated derivation , then , for all .

Proof. We prove only (ii), since (i) is proved in [2]. For all we have and . Comparing the two expressions for gives .

Lemma 6. Let be a 3-prime near ring and a generalized derivation with associated derivation .(i) for all .(ii) for all .

Proof. (i) , and . Comparing these two equations gives the desired result.
(ii) Again, calculate and and compare.

Lemma 7. Let be a 3-prime near ring and a generalized derivation with associated derivation . Then .

Proof. Let and . Then ; that is, . Applying Lemma 6(i), we get . It follows that for all , so .

Lemma 8. Let be a 3-prime near ring and a nonzero semigroup ideal of . If is a nonzero right generalized derivation of with associated derivation , then .

Proof. Suppose . Then for all and , and it follows by Lemma 3(ii) that . Therefore for all and , and another appeal to Lemma 3(ii) gives , which is a contradiction.

Lemma 9 (see [2, Theorem 4.2]). Let be a 3-prime near ring with a nonzero right generalized derivation with associated derivation . If then is abelian. Moreover, if is 2-torsion-free, then is a commutative ring.

Lemma 10 (see [2, Theorem 4.1]). Let be a 2-torsion-free 3-prime near ring and a nonzero generalized derivation on with associated derivation . If , then 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. Let be a 3-prime near ring and be a nonzero semigroup ideal of . Let be a nonzero right generalized derivation with associated derivation . If , then is abelian. Moreover, if is 2-torsion-free, then is a commutative ring.

Proof. We begin by showing that is abelian, which by Lemma 2(ii) is accomplished by producing such that . Let be an element of such that . Then for all , and , so that and ; hence we need only to show that there exists such that . Suppose that this is not the case, so that , for all . By Lemma 3(i) either or .
If , then , for all . Thus , for all , and . This implies that , for all and and Lemma 2(i) gives . Thus , for all . Replacing by , we have , and by Lemmas 2(i) and 8, we get that . Thus, we have a contradiction.
To complete the proof, we show that if is 2-torsion-free, then is commutative.
Consider first the case . This implies that for all and . By Lemma 8, we have such that , so is commutative by Lemma 2(iii).
Now consider the case . Let . This implies that if , . Thus for all and . Therefore by Lemma 5(i), for all and . Since and , we obtain , for all and . Let . Choosing such that and noting that is not a zero divisor, we have for all . By Lemma 1(ii), ; hence is commutative by Lemma 4.
The remaining case is and . Suppose we can show that . Taking and , we have ; therefore by Lemma 2(iii) and is commutative by Lemma 9.
Assume, then, that . For each , , so . Thus, for all and , , so , and by Lemma 3(iii)  . Since for all and , we have , and right multiplying by gives . Consequently , so that for all . Since , for all , so for all . But by Lemma 8, there exists for which ; and since , we get —a contradiction. Therefore as required.

Theorem 12. Let be a 3-prime near ring with a nonzero generalized derivation with associated nonzero derivation . Let be a nonzero semigroup ideal of . If , then is abelian.

Proof. Assume that is such that . For all such that , .
This implies that This equation may be restated as , where .
Let . Then and , so , and by the argument in the previous paragraph, . We now have for all such that . Taking and , we get , and since is a nonzero semigroup ideal by Lemma 3(ii) and , Lemma 3(i) gives Take and , where and , so that . By (2) we have Replacing by , , we obtain ; so by (3) for all and . It follows immediately by Lemmas 1(i) and 3(i) that for all ; that is, is abelian.

Theorem 13. Let be a 2-torsion-free 3-prime near ring and be a nonzero semigroup ideal of . If is a nonzero generalized derivation with associated derivation such that , then is a commutative ring if it satisfies one of the following: (i)  ; (ii) ; (iii) and .

Proof. (i) Let centralizes , and let such that . Then a centralizes for all , so that and . Since , for all . Therefore a centralizes , and by Lemma 1(ii), . Since centralizes , and our result follows by Theorem 11.
(ii) We may assume . Let . Then for all , and commute; hence . Our result now follows from Lemma 10.
(iii) Let and such that . Then , and since , . Thus centralizes , and by Lemma 1(ii), . Our result now follows by Theorem 11.

We have already observed that if is a generalized derivation with , then for all . For 3-prime near rings, we have the following converse.

Theorem 14. Let be a 3-prime near ring and be a nonzero semigroup ideal of . If is a nonzero right generalized derivation of with associated derivation and , for all , then .

Proof. We are given that for all . Substituting for , we get for all . It follows that for all ; that is, for all . By Lemma 3(i), , and hence by Lemma 1(i).

4. Generalized Derivations Acting as a Homomorphism or an Antihomomorphism

In [4], Bell and Kappe proved that if is a semiprime ring and is a derivation on which is either an endomorphism or an antiendomorphism on , then . Of course, derivations which are not endomorphisms or antiendomorphisms on may behave as such on certain subsets of ; for example, any derivation behaves as the zero endomorphism on the subring consisting of all constants (i.e., the elements for which ). In fact in a semiprime ring , may behave as an endomorphism on a proper ideal of . However as noted in [4], the behaviour of is somewhat restricted in the case of a prime ring. Recently the authors in [6] considered -derivation acting as a homomorphism or an antihomomorphism on a nonzero Lie ideal of a prime ring and concluded that . 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. Let be a 3-prime near ring and be a nonzero semigroup ideal of . Let be a nonzero generalized derivation on with associated derivation . If acts as a homomorphism on , then is the identity map on and .

Proof. By the hypothesis Replacing by in the above relation, we get or This implies that Using Lemma 5(ii), we get or This implies that That is, Therefore which implies that It follows by Lemma 3(i) that either or for all .
In fact, as we now show, both of these conditions hold.
Suppose that for all . Then for all and , ; hence for all , and thus .
On the other hand, suppose that , so that . Then for all , , so that . Replacing by , , and noting that , we see that for all . Therefore, or is the identity on . But contradicts Lemma 8, so is the identity on .
We now know that is the identity on and for all . Consequently, for all and , so that for all . It follows that is the identity on .

Theorem 16. Let be a 3-prime near ring and be a nonzero semigroup ideal of . If is a nonzero generalized derivation on with associated derivation . If acts as an antihomomorphism on , then , is the identity map on , and is a commutative ring.

Proof. We begin by showing that if and only if is the identity map on .
Clearly if is the identity map on , for all , and hence .
Conversely, assume that , in which case for all . It follows that for any , On the other hand, Comparing (14) and (15) shows that centralizes , so that by Lemma 1(ii).
Now is a nonzero semigroup ideal by Lemma 3(ii); hence by Lemma 8. Choosing such that , we see that for any , , and hence . Since , we conclude that for all , and it follows easily that is the identity map on .
We note now that if the identity map on acts as an antihomomorphism on , then is commutative, so that by Lemmas 1(ii) and 4 is a commutative ring.
To complete the proof of our theorem, we need only to argue that . By our antihomomorphism hypothesis Replacing by in the above relation, we get This implies that Using Lemma 5(ii), we get Thus Replacing by in (20) and using (20), we get Application of Lemma 3(i) yields that for each either or ; that is or .
Suppose that there exists such that . Then for all such that , , and hence . Now consider arbitrary . If one of is in , then . If , then , so . Therefore for all , and by Theorem 15, is the identity map on , and therefore .
The remaining possibility is that for each , either or . Let , and let . Then is a nonzero semigroup right ideal contained in and is an additive subgroup of . The sets and are additive subgroups of with union equal to , so or . If , then by Lemma 1(i). Suppose, then, that . Then for arbitrary   , so , and again . 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.

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