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International Journal of Mathematics and Mathematical Sciences
Volume 2010 (2010), Article ID 646587, 6 pages
http://dx.doi.org/10.1155/2010/646587
Research Article

Remarks on Generalized Derivations in Prime and Semiprime Rings

Department of Mathematics, Belda College, Belda, Paschim Medinipur 721424, India

Received 15 August 2010; Accepted 28 November 2010

Academic Editor: Hans Keiding

Copyright © 2010 Basudeb Dhara. 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 ring with center and a nonzero ideal of . An additive mapping is called a generalized derivation of if there exists a derivation such that for all . In the present paper, we prove that if for all or for all , then the semiprime ring must contains a nonzero central ideal, provided . In case is prime ring, must be commutative, provided . The cases (i) and (ii) for all are also studied.

1. Introduction

Let be an associative ring. The center of is denoted by . For , the symbol will denote the commutator and the symbol will denote the anticommutator . We will make extensive use of basic commutator identities , . An additive mapping from to is called a derivation of if holds for all . An additive mapping from to is called a generalized derivation of if there exists a derivation from to such that holds for all . Obviously, every derivation is a generalized derivation of . Thus, generalized derivation covers both the concept of derivation and left multiplier mapping. A mapping from to is called centralizing on where , if for all .

Over the last several years, a number of authors studied the commutativity in prime and semiprime rings admitting derivations and generalized derivations. In [1], Daif and Bell proved that if is a semiprime ring with a nonzero ideal and is a derivation of such that for all , then is central ideal. In particular, if , then is commutative. Recently, Quadri et al. [2] generalized this result replacing derivation with a generalized derivation in a prime ring . More precisely, they proved the following.

Let be a prime ring and a nonzero ideal of . If admits a generalized derivation associated with a nonzero derivation such that any one of the following holds: (i)    for all , (ii)    for all , (iii)    for all ; (iv) for all , then is commutative.

In the present paper, we study all these cases in semiprime ring.

2. Main Results

We recall some known results on prime and semiprime rings.

Lemma 2.1 (see [3, Lemma 1.1.5]or [1, Lemma 2]). (a) If is a semiprime ring, the center of a nonzero one-sided ideal is contained in the center of , in particular, any commutative one-sided ideal is contained in the center of .
(b) If is a prime ring with a nonzero central ideal, then must be commutative.

Lemma 2.2 (see [1, Lemma 1]). Let be a semiprime ring and a nonzero ideal of . If and centralizes , then centralizes .

Lemma 2.3 (see [4, Theorem 3]). Let be a semiprime ring and a nonzero left ideal of . If admits a derivation which is nonzero on and centralizing on , then contains a nonzero central ideal.

Now we begin with the theorem.

Theorem 2.4. Let be a semiprime ring, a nonzero ideal of and a generalized derivation of associated with a derivation of such that . If for all , then contains a nonzero central ideal.

Proof. By our assumption, we have that for all . If , then we find that for all , that is, is commutative. Then, by Lemma 2.1, and thus we obtain our conclusion.
Next assume that . Putting in (2.1), we get that Since is a generalized derivation of associated with a derivation of , (2.2) gives Using (2.1), it reduces to for all . Now putting in (2.4), we get Using (2.4), it gives for all . Now we put in (2.6) and obtain that for all . Right multiplying (2.6) by and then subtracting from (2.7), we get for all . This implies for all that and so , forcing . Then by Lemma 2.3, contains a nonzero central ideal.

Corollary 2.5. Let be a prime ring, a nonzero ideal of and a generalized derivation of . If for all , then is commutative or for all .

Proof. Let be the associated derivation of . By Theorem 2.4, we conclude that either or is commutative. Assume that is not commutative. Then . Since is a prime ring, implies and hence for all . Set for all . Then for all . Now, our assumption gives , that is, for all . Thus using , we have , that is, for all . Thus . Since is prime, this implies or is commutative. By Lemma 2.1, commutative implies that is commutative, a contradiction. Thus which gives for all .

Theorem 2.6. Let be a semiprime ring, a nonzero ideal of and a generalized derivation of associated with a derivation of such that . If for all , then contains a nonzero central ideal.

Proof. If , then by our assumption we have that , that is, for all . This implies that for all and so , forcing . Therefore, for all , gives , that is, is commutative. Then by Lemma 2.1, and thus we obtain our conclusion.

Next assume that . Then for any , we have Since is a generalized derivation associated with a derivation , above expression yields Putting in (2.10), we have Right multiplying (2.10) by and then subtracting from (2.11), we get for all . Replacing with in (2.12) and then again using (2.12) we find that Again replacing with in (2.13) and then using (2.13) we obtain for all , which is the same identity as (2.8) in the proof of Theorem 2.4. Thus by the same argument as in the proof of Theorem 2.4, we conclude that contains a nonzero central ideal.

Corollary 2.7. Let be a prime ring, a nonzero ideal of and a generalized derivation of . If for all , then is commutative or for all .

Proof. Let be the associated derivation of . By Theorem 2.6, we conclude that either or is commutative. If is not commutative, then . Since is a prime ring, implies and hence for all . Set for all . Then for all . Now, our assumption gives , that is, for all . Thus using , we have , that is, for all . Thus . Since is prime, this implies or is commutative. By Lemma 2.1, commutative implies that is commutative, a contradiction. Therefore, and hence for all .

Theorem 2.8. Let be a semiprime ring with center , a nonzero ideal of and a generalized derivation of associated with a derivation of . If for all , then .

Proof. We have for all . Since , we may choose . Then for any . Now we replace with in (2.15) and then we get By (2.15), we have for all . Since , this gives that for any , which implies for all . By Lemma 2.2, for all . Since , this gives for all and for all . Thus, , that is, .

Corollary 2.9. Let be a prime ring with center , a nonzero ideal of and a generalized derivation of associated with a derivation . If and for all , then is commutative.

Proof. Since and contains no nonzero elements which are zero divisors, we have from Theorem 2.8 that . Then by Lemma 2.1(b), we obtain our conclusion.

Theorem 2.10. Let be a semiprime ring with center , a nonzero ideal of and a generalized derivation of associated with a derivation of . If for all , then .

Proof. We have for all . Since , we choose . Then for any . Now we replace with in (2.17) and then we get By (2.17), we have that is for all . Now putting and , , respectively, we obtain that and . Subtracting these two results yields for all and for all . This gives for all and for all . We know the Jacobian identity for any . Using this identity, it follows that By using (2.19), it reduces to for all and for all . By Lemma 2.2, this implies that , that is, . Thus and then again by Lemma 2.2, . This yields which implies , since . Since is any nonzero element in , we conclude that .

Acknowledgment

The author would like to thank the referees for providing very helpful comments and suggestions.

References

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