ISRN Algebra

Volume 2014 (2014), Article ID 684792, 8 pages

http://dx.doi.org/10.1155/2014/684792

Research Article

On Generalized Jordan Triple -Higher Derivations in Prime Rings

Mohammad Ashraf and Almas Khan

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

Received 25 October 2013; Accepted 26 November 2013; Published 22 January 2014

Academic Editors: V. De Filippis, T. Nakatsu, and U. Vishne

Copyright © 2014 Mohammad Ashraf and Almas Khan. 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 and let be a Lie ideal of . Suppose that are endomorphisms of , and is the set of all nonnegative integers. A family of mappings is said to be a generalized -higher derivation (resp., generalized Jordan triple -higher derivation) of if there exists a -higher derivation of such that , the identity map on , , and (resp., hold for all and for every If the above conditions hold for all , then is said to be a generalized -higher derivation (resp., generalized Jordan triple -higher derivation) of into . In the present paper it is shown that if is a noncentral square closed Lie ideal of a prime ring of characteristic different from two, then every generalized Jordan triple -higher derivation of into is a generalized -higher derivation of into .

1. Introduction

Let be an associative ring with center (may be without identity element). For any , will denote the usual Lie product and the element will be denoted by . An additive subgroup of is said to be a Lie ideal of if . A Lie ideal of is said to be square closed Lie ideal of if for all . An additive mapping is said to be a derivation on if for all . Also, an additive mapping is called a Jordan triple derivation if for all . Herstein [1, Lemma 3.5] showed that on a 2-torsion free ring every derivation is Jordan triple derivation, but the converse need not be true in general, whereas Brešar [2] proved that on a 2-torsion free semiprime ring every Jordan triple derivation is a derivation.

The concept of derivations was further extended to -derivation. Let be the endomorphisms of . An additive mapping is said to be a -derivation of if holds for all . Moreover, an additive mapping is said to be a Jordan triple -derivation of if holds for all . Brešar [3] introduced the notion of generalized derivation as follows: an additive mapping is said to be a generalized derivation on if there exists a derivation on such that for all . An additive mapping is said to be a generalized -derivation on if there exists a -derivation on such that for all (for reference see [4]). Correspondingly, the concept of generalized Jordan triple derivations was defined by Wu and Lu [5]. An additive mapping is said to be a generalized Jordan triple derivation on if there exists a Jordan triple derivation on such that for all . They also showed that every generalized Jordan triple derivation on a 2-torsion free prime ring is a generalized derivation. Further, Liu and Shiue [6] extended this result for generalized Jordan triple -derivation. An additive mapping is said to be a generalized Jordan triple -derivation on if there exists a Jordan triple -derivation on such that for all . Liu and Shiue proved that on a 2-torsion free semiprime ring every generalized Jordan triple -derivation is a generalized -derivation.

Various results proved for derivations and generalized derivations were shown to be true in case of higher derivations which can be found in [7–10], and so forth. Let be the set of all nonnegative integers. Following Hasse and Schmidt [11], a family of additive mappings on is said to be a higher derivation on if   (the identity map on ) and holds for all and for each .

The concept of higher derivation was extended to -higher derivation by the authors together with Haetinger [7]. Let be a family of maps . Then is said to be a -higher derivation on   if  , and   hold for all and for each . If is a Lie ideal of , then is said to be a -higher derivation of into if the above conditions hold for all . Further, in [8] the authors introduced the notion of generalized -higher derivation in rings. A family of mappings is said to be a generalized -higher derivation of if there exists a -higher derivation of such that , and for all and for each .

Motivated by the concepts of Jordan triple derivation, generalized derivation, and -higher derivation, we introduce generalized Jordan triple -higher derivation as follows: a family of mappings is said to be a generalized Jordan triple -higher derivation of if there exists a -higher derivation of such that and   hold for all   and every . Further, if is a -higher derivation of into and the above conditions hold for all , then is said to be a generalized Jordan triple -higher derivation of into .

It can be easily seen that, on a 2-torsion free ring , every generalized -higher derivation of a square closed Lie ideal into is a generalized Jordan triple -higher derivation of into . In fact, if is generalized -higher derivation of a square closed Lie ideal into , then for all and for each . Replacing with and using the fact that is 2-torsion free, we obtain for all and for each   ; that is, is generalized Jordan triple -higher derivation of   into but the converse need not be true in general.

In the present paper, our objective is to find the conditions on under which every generalized Jordan triple -higher derivation of   into is a generalized -higher derivation of   into . In fact our result generalizes the main theorem obtained in [4, 7–9, 12–14].

2. Main Results

Throughout, will denote the endomorphisms of such that is one-one and onto and . Note that if a Lie ideal   of is square closed, then for all and since ; yields . Let be a generalized Jordan triple -higher derivation of   into with associated -higher derivation of into . For every fixed and each , we denote by the element of   as

Then, by using similar arguments as used in Lemma 2.2(iii) of [8], it can easily be seen that is additive in each argument and .

Following lemmas are essential for developing the proof of our main result. Proof of Lemma 1 can be seen in [15] whereas Lemma 2 was obtained in [9].

Lemma 1. Assume that is a prime ring of characteristic different from two and is a noncentral square closed Lie ideal of . Let be additive groups, and let be the mappings which are additive in each argument. If for every , , , then for every , or for every , .

Lemma 2. Let be a prime ring of characteristic different from two and let be a square closed Lie ideal of such that . Then there exist elements such that .

Now, we prove the following lemma which plays a crucial role in developing the proof of our main result.

Lemma 3. Let be a 2-torsion free ring and let be a square closed Lie ideal of . If is a generalized Jordan triple -higher derivation of into and for all and every , then for all and every .

Proof. Let . Then

From our hypothesis we have

Linearizing the above equation we obtain

Again consider . Using relation (4) we have

Combining the two equations (2) and (5)

Considering the first term,

Calculating the second term,

Using the hypothesis that for all , we have

Now, subtracting the two terms so obtained we find that

Similarly, the difference of the last two terms yields

Thus, (6) reduces to

Since the characteristic of is not two, we find that .

Lemma 4. Let be a prime ring of characteristic different from 2 and let be a noncentral square closed Lie ideal of . If is a generalized Jordan triple -higher derivation on into , then , for all and each .

Proof. Trivially for , . By induction let us assume that , for every and for all . By Lemma 3, we have or . Again, by Lemma 1, we have for all or for all . If for all holds, then this contradicts Lemma 2. Hence, in either case for all .

Theorem 5. Let be a prime ring of characteristic different from 2 and let be a square closed Lie ideal of such that . Then, every generalized Jordan triple -higher derivation of into is a generalized -higher derivation of into .

Proof. Let be generalized Jordan triple -higher derivation.

It can be easily seen that . By induction, assume that holds for all .

For , take and use Lemma 4 to get

On the other hand,

Comparing both equations, reordering the indices and as characteristic of is different from two we finally obtain

As is one-one and onto implementing Lemma 1, it can be easily seen that for all or for all . Now if for all , then is commutative and hence using similar arguments as used in the second paragraph of the proof of Lemma 1.3 of [1] is central, a contradiction. Therefore, for all . This completes the proof of our theorem.

Remark 6. For and for each our result reduces to the main result obtained by Ferrero and Haetinger [9].

Corollary 7 ([9, Theorem 1.2]). Let be a prime ring of characteristic different from two and let be a noncentral square closed Lie ideal of . Then, every Jordan triple higher derivation of into is a higher derivation of into .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors wish to thank the referees for their useful suggestions.

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