International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 625968, 14 pages

doi:10.1155/2012/625968

Research Article

On Prime-Gamma-Near-Rings with Generalized Derivations

Kalyan Kumar Dey,¹ Akhil Chandra Paul,¹ and Isamiddin S. Rakhimov²

¹Department of Mathematics, Rajshahi University, Rajshahi 6205, Bangladesh

²Department of Mathematics, Faculty of Science and Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, 43400 Serdang, Malaysia

Received 26 January 2012; Accepted 19 March 2012

Academic Editor: B. N. Mandal

Copyright © 2012 Kalyan Kumar Dey 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 2-torsion free prime -near-ring with center . Let and be two generalized derivations on . We prove the following results: (i) if or or for all , , then is a commutative -ring. (ii) If and for all , , then . (iii) If acts as a generalized derivation on , then or .

1. Introduction

The derivations in -near-rings have been introduced by Bell and Mason [1]. They studied basic properties of derivations in -near-rings. Then Aşci [2] obtained commutativity conditions for a -near-ring with derivations. Some characterizations of -near-rings and regularity conditions were obtained by Cho [3]. Kazaz and Alkan [4] introduced the notion of two-sided --derivation of a -near-ring and investigated the commutativity of a prime and semiprime -near-rings. Uçkun et al. [5] worked on prime -near-rings with derivations and they found conditions for a -near-ring to be commutative. In [6] Dey et al. studied commutativity of prime -near-ring with generalized derivations.

In this paper, we obtain the conditions of a prime -near-ring to be a commutative -ring. If , and for all , then is central. Also we prove that if is the generalized derivation on , then and are trivial.

2. Preliminaries

A -near-ring is a triple , where

(i) is a group (not necessarily abelian);(ii) is a nonempty set of binary operations on such that for each is a left near-ring;(iii), for all and .

We will use the word -near-ring to mean left -near-ring. For a near-ring , the set is called the zero-symmetric part of . A -near-ring is said to be zero-symmetric if . Throughout this paper, will denote a zero symmetric left -near-ring with multiplicative centre . Recall that a -near-ring is prime if implies or . An additive mapping is said to be a derivation on if for all , or equivalently, as noted in [1], that for all . Further, an element for which is called a constant. For , the symbol will denote the commutator , while the symbol will denote the additive-group commutator . An additive mapping is called a generalized derivation if there exits a derivation of such that for all . The concept of generalized derivation covers also the concept of a derivation.

3. Derivations on Γ-Near-Rings

In this section we prove that a few subsidiary results (Lemmas 3.1, 3.2, 3.4, 3.8, 3.9, 3.10 and 3.11) to use them for proving of our main results (Theorems 3.3, 3.5, 3.6, 3.12 and 3.13).

Lemma 3.1. Let be an arbitrary derivation on a -near-ring . Then satisfies the following partial distributive law: and for all .

Proof. For all , we get and . Equating these two relations for now yields the required partial distributive law.

Lemma 3.2. Let be a derivation on a -near-ring and suppose is not a left zero divisor. If , then is a constant for every .

Proof. From , for all , , we obtain , which reduces , for all .

Since , , this equation is expressible as . Thus .

Theorem 3.3. Let be a -near-ring having no nonzero divisors of zero. If admits a nontrivial commuting derivation , then is abelian.

Proof. Let be any additive commutator. Then is a constant by Lemma 3.2. Moreover, for any , , is an additive commutator, hence also a constant. Thus, and , for all . Since for all , we conclude that .

Lemma 3.4. Let be a prime -near-ring.

(i)If , then is not a zero divisor in .(ii)If contains an element for which , then is abelian.(iii)Let be a nonzero derivation on . Then implies , and implies .(iv)If is 2-torsion free and is a derivation on such that , then .

Proof. (i) If and , , , then , , , . Thus we get , by primeness of , .

(ii) Let be an element such that , and let . Since is distributive, we get .

On the other hand, . Thus, and therefore . Hence is abelian.

(iii) Let , and let be arbitrary elements of and . Then . Thus , and since .

A similar argument works if , since Lemma 3.1 provides enough distributivity to carry it through.

(iv) For arbitrary , we have . Since is 2-torsion free, . Thus for each , and (iii) yields . Thus .

Theorem 3.5. If a prime -near-ring admits a nontrivial derivation for which , then is abelian. Moreover, if is 2-torsion free, then is a commutative -ring.

Proof. Let be an arbitrary constant, and let be anon-constant. Then , . Since , it follows easily that . Since is a constant for all constants , it follows from Lemma 3.4(ii) that is abelian, provided that there exists a nonzero constant.

Assume, then, that 0 is the only constant. Since is obviously commuting, it follows from Lemma 3.2 that all which are not zero divisors belong to the center of , denoted by . In particular, if , . But then for all , , hence .

Now we assume that is 2-torsion free. By Lemma 3.1, for all , and using the fact that , , we get . Since is abelian and , this equation can be rearranged to yield for all .

Suppose now that is not commutative. Choosing , with , , and letting , we get , for all , and since the central elements cannot be a nonzero divisor of zero, we conclude that for all . But by Lemma 3.4(iv), this cannot happen for nontrivial .

Theorem 3.6. Let be a prime -near-ring admitting a nontrivial derivation such that for all . Then is abelian. Moreover, if is 2-torsion free, then is a commutative -ring.

Proof. By Lemma 3.4(ii), if both and commute element-wise with , then , for all additive commutators . Thus, taking , we get for all , so by Lemma 3.4(iii). Since is also an additive commutator for any , , we have , and another application of Lemma 3.4(iii) gives .

Now we assume that is 2-torsion free. By the partial distributive law, for all , hence, . Thus for all , .

Replacing , we obtain , for all ,, so that for all , . The primeness of shows that either or , and since the first of these conditions is impossible by Lemma 3.4(iv), the second must hold and is a commutative -ring by Theorem 3.5.

Definition 3.7. Let be a -near-ring and a derivation of . An additive mapping is said to be a right generalized derivation of associated with if

and is said to be a left generalized derivation of associated with if

Finally, is said to be a generalized derivation of associated with if it is both a left and right generalized derivation of associated with .

Lemma 3.8. Let be a right generalized derivation of a -near ring associated with . Then

(i) for all ;(ii) for all .

Proof. (i) For any , we get

Comparing these two expressions, we obtain

and so,

(ii) In a similar way.

Lemma 3.9. Let be a right generalized derivation of a -near ring associated with . Then

(i), for all , .(ii), for all , .

Proof. (i) For any , , we get .

On the other hand,

From these two expressions of , we obtain that, for all , ,

(ii) The proof is similar.

Lemma 3.10. Let be a prime -near-ring, a nonzero generalized derivation of associated with the nonzero derivation and . (i) If , then . (ii) If , then .

Proof. (i) For any , we get . Hence . Since is a prime -near-ring and , we obtain .

(ii) A similar argument works if .

Lemma 3.11. Let be a prime -near-ring. Let be a generalized derivation of associated with the nonzero derivation . If is a 2-torsion free -near-ring and , then .

Proof. (i) For any , , we get

By the hypothesis,

Writing by in (3.9), we get , for all .

By Lemma 3.9(ii), we obtain that or . If then from Lemma 3.4(iv), a contradiction. So we find .

Theorem 3.12. Let be a prime -near-ring with a nonzero generalized derivation associated with . If , then is abelian. Moreover, if is 2-torsion free, then is commutative -ring.

Proof. Suppose that , such that . So, and . For all , , we have .

That is, , for all , .

Since , we get , and so, for all .

Since and is a prime -near-ring, it follows that , for all . Thus is abelian.

Using the hypothesis, for any , . By Lemma 3.4(ii), we have . Using and being abelian, we obtain that

Substituting for in (3.10), we get for all .

Since and f a nonzero generalized derivation with associated with , we get . So, is a commutative -ring by Theorem 3.3.

Theorem 3.13. Let be a prime -near-ring with a nonzero generalized derivation associated with . If , , then is abelian. Moreover, if is 2-torsion free, then is commutative -ring.

Proof. By the same argument as in Theorem 3.12, it is shown that if both and commute elementwise with , then we have

Substituting for in (3.11), we get , . By Lemma 3.9(i), we obtain that for all , . For any , , we have and so, we obtain , for any , . From Lemma 3.4(iii), we get for any .

Now we assume that is 2-torsion free. By the assumption , , we have

Hence we get

and so,

If we take instead of in (3.14), then

and so,

Thus we get , for all , . Since is a prime -near-ring, we have or . Let us assume that . Then

and so,

Replacing by , , in (3.18), we get

Using (3.18) and being 2-torsion free -near-ring, we get .

Thus we obtain that . It contradicts by . The theorem is proved.

4. Generalized Derivations of Γ-Near-Rings

We denote a generalized derivation determined by a derivation of by . We assume that is a nonzero derivation of unless stated otherwise.

Theorem 4.1. Let be a generalized derivation of . If   for all , then is a commutative -ring.

Proof. Assume that for all , . Substitute instead of , obtaining

Since the second term is zero, it is clear that

Replacing by in (4.2) and using this equation, we get

Hence either or . Let . Then and are two additive subgroups of . However, a group cannot be the union of proper subgroups, hence either or . Since , we are forced to conclude that is a commutative -ring.

Theorem 4.2. Let be a generalized derivation of . If   for all , , then is a commutative -ring.

Proof. Assume that for all , . Replacing by , , in the hypothesis, we have

On the other hand,

It follows from the two expressions for that

Using the same argument as in the proof of Theorem 4.1, we get that is a commutative -ring.

Theorem 4.3. Let be a nonzero generalized derivation of . If f acts as a homomorphism on , then is the identity map.

Proof. Assume that acts as a homomorphism on . Then one obtains

Replacing by in (4.7), we arrive at

Since is a generalized derivation and acts as a homomorphism on , we deduce that

By Lemma 3.9(ii), we get

and so

That is,

Hence, we deduce that

Because is prime and , we have for all . Thus, is the identity map.

Theorem 4.4. Let be a nonzero generalized derivation of . If f acts as an antihomomorphism on , then is the identity map.

Proof. By the hypothesis, we have

Replacing by in the last equation, we obtain

Since is a generalized derivation and acts as an antihomomorphism on , we get

By Lemma 3.9(ii), we conclude that

and so

Replacing by and using this equation, we have

Hence we obtain the following alternatives: or , for all . By a standard argument, one of these must hold for all . Since , the second possibility gives that is commutative -ring by Theorem 3.12, and so we deduce that is the identity map by Theorem 4.3.

Theorem 4.5. Let be a generalized derivation of such that , and . If   for all , , then .

Proof. Since , there exists such that . Furthermore, as is a derivation, it is clear that . Replacing by , , in the hypothesis and using Lemma 3.9(ii), we have

Since and , we get

By the primeness of and , we obtain that .

Theorem 4.6. Let be a generalized derivation of , and . If   for all , then .

Proof. If , then there is nothing to prove. Hence, we assume that .

Replacing by in the hypothesis, we have

Using , we have

Taking instead of in the last equation and using this, we conclude that

Since is a prime -near-ring, we have either or . If , then is abelian by Lemma 3.2(ii). Thus

and so

That is, . Hence in either case we have . This completes the proof.

Theorem 4.7. Let be a generalized derivation of . If is a 2-torsion free -near-ring and , then is a commutative -ring.

Proof. Suppose that . Then we get

In particular, for all , , . Since the first summand is an element of , we have

Taking instead of in (4.28), we obtain that

Since , , and so for all , , , we conclude for all , , .

Since is prime, we get or . If , then is a commutative -ring by Lemma 3.8. Hence, we assume . By (4.28), we get for all , , .

Since is a 2-torsion free near-ring and , we obtain that either or . If , then we are already done. So, we may assume that . Then

and so

Now replacing by in (4.31), and using the fact that , we get

That is,

Again taking instead of in this equation, one can obtain

The second term is equal to zero because of . Hence we have for all , , .

Since by the hypothesis, we get either or . If , then the theorem holds by Definition 3.7. If , then for all , , and so

Using , we now have for all , , . Since , we have either or . If , then is a derivation of and so is commutative -ring by Lemma 3.11.

Now assume that . Returning to the equation (4.31), we have

Since , we have either or . Clearly, implies the theorem holds. If , then by the hypothesis, and so is a commutative -ring by Lemma 3.4(iv). Hence, the proof is completed.

Corollary 4.8. Let be a 2-torsion free near-ring, and a nonzero generalized derivation of . If , , then is a commutative -ring.

Lemma 4.9. Let and be two generalized derivations of . If h is a nonzero derivation on and for all , then is abelian.

Proof. Suppose that for all , .

We substitute for , thereby obtaining

Using the hypothesis, we get

It follows by Lemma 3.10(ii) that for all . For any , we have and so for all , .

An appeal to Lemma 3.4(iii) yields that is abelian.

Theorem 4.10. Let and be two generalized derivations of . If is 2-torsion free and for all , then or .

Proof. If or , then the proof of the theorem is obvious. So, we may assume that and . Therefore, we know that is abelian by Lemma 4.9.

Now suppose that

Replacing by in this equation and using the hypothesis, we get

and so

Taking instead of in the above relation, we obtain

That is,

Replacing by in (4.43) and using this relation, we have

Since is a prime -near-ring and , we obtain that

Now again taking instead of in the initial hypothesis, we get

Using (4.45) yields that

Taking instead of in this equation, we arrive at

By the hypothesis and (4.45), we have

and so

Since is a 2-torsion free prime -near-ring, we obtain that . An appeal to Lemma 3.4(iii) and (iv) gives that . This contradicts by our assumption. Thus the proof is completed.

Theorem 4.11. Let and be two generalized derivations of . If acts as a generalized derivation on , then or .

Proof. By calculating in two different ways, we see that for all , . The proof is completed by using Theorem 4.10.

Acknowledgments

The paper was supported by Grant 01-12-10-978FR MOHE Malaysia. The authors are thankful to the referee for valuable comments.

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