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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 270132, 14 pages
http://dx.doi.org/10.1155/2012/270132
Research Article

Generalized Derivations in Semiprime Gamma Rings

1Department of Mathematics, University of Rajshahi, Rajshahi 6205, Bangladesh
2Department of Mathematics, FS, and Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, 43400 Serdang, Malaysia

Received 10 November 2011; Revised 5 December 2011; Accepted 5 December 2011

Academic Editor: Christian Corda

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 semiprime -ring satisfying the condition for all , and let be an additive mapping such that for all and for some derivation of . We prove that is a generalized derivation.

1. Introduction

Hvala [1] first introduced the generalized derivations in rings and obtained some remarkable results in classical rings. Daif and Tammam El-Sayiad [2] studied the generalized derivations in semiprime rings. The authors consider an additive mapping of a ring with the property for some derivation of . They prove that is a Jordan generalized derivation.

Aydin [3] studied generalized derivations of prime rings. The author proved that if is an ideal of a noncommutative prime ring , is a fixed element of and is an generalized derivation on associated with a derivation then the condition or for all implies .

Çeven and Öztürk [4] have dealt with Jordan generalized derivations in Γ-rings and they proved that every Jordan generalized derivation on some Γ-rings is a generalized derivation.

Generalized derivations of semiprime rings have been treated by Ali and Chaudhry [5]. The authors proved that if is a commuting generalized derivation of a semiprime ring associated with a derivation then for all and is central. They characterized a decomposition of relative to the generalized derivations.

Atteya [6] proved that if is nonzero ideal of a semiprime ring and admits a generalized derivation such that then contains a nonzero central ideal.

Rehman [7, 8] studied the commutativity of a ring by means of generalized derivations acting as homomorphisms and antihomomorphisms.

In this paper, we prove the following results

Let be a 2-torsion-free semiprime Γ-ring satisfying the following assumption: and be an additive mapping. If there exists a derivation of such that for all , , then is a Jordan generalized derivation.

2. Preliminaries

Let and Γ be additive abelian groups. is called a Γ-ring if there exists a mapping such that for all , the following conditions are satisfied:(i),(ii), , , .

For any and for the expressions is denoted by and are denoted by . Then one has the following identities: for all and for all . Using the assumption (*) the above identities reduce to for all and , .

Further stands for a prime Γ-ring with center . The ring is -torsion-free if , implies , where is a positive integer, is prime if implies or , and it is semiprime if implies . An additive mapping is called a left (right) centralizer if for , and it is called a Jordan left (right) centralizer if A mapping is called biadditive if it is additive in both arguments. An additive mapping is called a derivation if for all , and it is called a Jordan derivation if for all , . A derivation is inner if there exists , such that holds for all , . Every derivation is a Jordan derivation. The converse is in general not true. An additive mapping is said to be a generalized derivation if there exists a derivation such that for all , . The maps of the form where , are fixed elements in and for all called the generalized inner derivation. An additive mapping is said to be a Jordan generalized derivation if there exists a derivation such that for all , . Hence the concept of a generalized derivation covers both the concepts of a derivation and a left centralizers and the concept of a Jordan generalized derivation covers both the concepts of a Jordan derivation and a left Jordan centralizers. An example of a generalized derivation and a Jordan generalized derivation is given in [4].

3. Main Results

We start from the following subsidiary results.

Lemma 3.1. Let be a semiprime Γ-ring. If are such that for all , , then .

Proof. Let . Then By semiprimeness of with respect to , it follows that .

Lemma 3.2. Let be a semiprime Γ-ring and biadditive mappings. If for all , then for all , .

Proof. First we replace with in the relation , and use the biadditivity of the and . Then we have Then Hence by semiprimeness of with respect to .
Now we replace by and obtain the assertion of the lemma with the similar observation as above.

Lemma 3.3. Let be a semiprime Γ-ring satisfying the assumption (*) and be a fixed element of . If for all , , then .

Proof. First we calculate the following expressions using the assumption (*), Since for all , , we get . By the semiprimeness of we get for all . Hence .

Lemma 3.4. Let be a Γ-ring satisfying the condition (*) and be a Jordan generalized derivation with the associated derivation . Let and . Then(i). In particular, if is 2-torsion-free, then(ii),(iii).

Proof. (i) We have , for all , . Then replacing by , and following the series of implications below we get the result:
(ii) Replace by in the above relation (3.5), then we get, Using the assumption (*), we conclude that Again, replacing by in (3.5) Adding both sides , we get, Comparing (3.7) and (3.9) we obtain, Since is 2-torsion-free, it gives
(iii) Replace for in (3.12), we get,

Definition 3.5. Let be a Γ-ring and be a Jordan generalized derivation with the associated derivation . Define for all and .

Lemma 3.6. The function has the following properties:(i). (ii). (iii). (iv).

Proof. The results easily follow from Lemma 3.4(i).

Remark 3.7. is a generalized derivation if and only if for all , .

Theorem 3.8. Let be a 2-torsion-free semiprime Γ-ring satisfying the condition (*). Let and . Then(i) for all ,(ii).

Proof. (i) From Lemma 3.4(iii) we get We set Since Again From (3.15) and (3.16) we find,
(ii) According to Lemma 3.4(ii) we have, Replace by in (3.18) we find Similarly . Therefore,

Lemma 3.9. , for all , .

Proof. From Theorem 3.8(ii), we have . Therefore due to Lemma 3.4(ii) .

Theorem 3.10. Let be a 2-torsion-free semiprime Γ-ring satisfying the assumption (*) and be a Jordan generalized derivation with associated derivation on . Then is a generalized derivation.

Proof. In particular, , for all , . This gives Then . Now we will compute each side of this equality by using (3.15) and the above relation, So we get Moreover, So we get Comparing the results of (3.22) and (3.24) and using the above relations we obtain which gives where stands for .
On the other hand, we have Now we use (3.15) and the properties of , to derive which gives Moreover, So we obtain Comparing (3.31) and (3.33), we derive Finally using (3.31) we get . But . By the semiprimeness of , we have . Again by the primeness of , we get . The proof is complete.

It is clear that if we let the derivation to be the zero derivation in the above theorem, we get the following result.

Theorem 3.11. Let be a 2-torsion-free semiprime Γ-ring and be an additive mapping which satisfies for all , . Then is a left centralizer

Proof. We have If we replace by , we get By replacing with and using (3.5), we arrive at But on the other hand, Comparing (3.37) and (3.38) we obtain If we linearize (3.39) in , we get Now we shall compute in two different ways. If we use (3.39) we have But if we use (3.40) we have Comparing (3.41) and (3.42) and introducing a bi-additive mapping we arrive at Equality (3.36) can be rewritten in this notation as . Using this fact and (3.43) we obtain Using first Lemma 3.2 and then Lemma 3.1 we have Now fix some and using Lemma 3.3 we get .
In particular, , for all . This gives Therefore . Both sides of this equality will be computed in few steps using (3.36), Since , we obtain On the other hand, we also have Finally using (3.48) we arrive at , but this means that . Hence, which implies . The proof is complete.

Acknowledgments

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

References

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