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A Note on Jordan Triple Higher *-Derivations on Semiprime Rings
We introduce the following notion. Let be the set of all nonnegative integers and let be a family of additive mappings of a *-ring R such that ; D is called a Jordan higher *-derivation (resp., a Jordan higher *-derivation) of R if (resp., ) for all and each . It is shown that the notions of Jordan higher *-derivations and Jordan triple higher *-derivations on a 6-torsion free semiprime *-ring are coincident.
Let be an associative ring, for any . Recall that is prime if implies or and is semiprime if implies . Given an integer , is said to be -torsion free if, for implies . An additive mapping satisfying and for all is called an involution and is called a -.
An additive mapping is called a derivation if holds for all , and it is called a Jordan derivation if for all . Every derivation is obviously a Jordan derivation and the converse is in general not true [1, Example ]. An influential Herstein theorem  shows that any Jordan derivation on a 2-torsion free prime ring is a derivation. Later on, Brešar  has extended Herstein’s theorem to 2-torsion free semiprime rings. A Jordan triple derivation is an additive mapping satisfying for all . Any derivation is obviously a Jordan triple derivation. It is also easy to see that every Jordan derivation of a 2-torsion free ring is a Jordan triple derivation [4, Lemma 3.5]. Brešar  has proved that any Jordan triple derivation of a 2-torsion free semiprime ring is a derivation.
Let be a -ring. An additive mapping is called a -derivation if holds for all , and it is called a Jordan -derivation if holds for all . We might guess that any Jordan -derivation of a 2-torsion free prime -ring is a -derivation, but this is not the case. It has been proved in  that noncommutative prime -rings do not admit nontrivial -derivations. A Jordan triple -derivation is an additive mapping with the property for all . It could easily be seen that any Jordan -derivation on a 2-torsion free -ring is a Jordan triple -derivation [6, ]. Vukman  has proved that any Jordan triple -derivation on a 6-torsion free semiprime -ring is a Jordan -derivation.
Let be the set of all nonnegative integers and let be a family of additive mappings of a ring such that . Then is said to be a higher derivation (resp., a Jordan higher derivation) of if, for each (resp., ) holds for all . The concept of higher derivations was introduced by Hasse and Schmidt . This interesting notion of higher derivations has been studied in both commutative and noncommutative rings; see, for example, [9–13]. Clearly, every higher derivation is a Jordan higher derivation. Ferrero and Haetinger  have extended Herstein's theorem  for higher derivations on 2-torsion free semiprime rings. For an account of higher and Jordan higher derivations the reader is referred to . A family of additive mappings of a ring , where , is called a Jordan triple higher derivation if holds for all . Ferrero and Haetinger  have proved that every Jordan higher derivation of a 2-torsion free ring is a Jordan triple higher derivation. They also have proved that every Jordan triple higher derivation of a 2-torsion free semiprime ring is a higher derivation.
Motivated by the notions of -derivations and higher derivations, we naturally introduce the notions of higher -derivations, Jordan higher -derivations, and Jordan triple higher -derivations. Our main objective in this paper is to show that every Jordan triple higher -derivation of a 6-torsion free semiprime -ring is a Jordan higher -derivation. This result extends the main result of . It is also shown that every Jordan higher -derivation of a 2-torsion free -ring is a Jordan triple higher -derivation. So we can conclude that the notions of Jordan triple higher -derivations and Jordan higher -derivations are coincident on 6-torsion free semiprime -rings.
2. Preliminaries and Main Results
We begin by the following definition.
Definition 1. Let be the set of all nonnegative integers and let be a family of additive mappings of a -ring such that . is called(a)a higher -derivation of if, for each , (b)a Jordan higher -derivation of if, for each , (c)a Jordan triple higher -derivation of if, for each ,
Throughout this section, we will use the following notation.
Notation. Let be a Jordan triple higher -derivation of a -ring . For every fixed and each , we denote by and the elements of defined by
It can easily be seen that , , and for each pair . We will use these relations without any explicit mention in the steps of the proofs. The next lemmas are crucial in developing the proofs of the main results.
Lemma 2 (see [5, Lemma 1.1]). Let be a 2-torsion free semiprime ring. If are such that for all , then for all . If is semiprime, then for all implies .
Lemma 3 (see [7, Lemma 1]). Let be a 2-torsion free semiprime -ring. If are such that for all , then .
Lemma 4. Let be a Jordan triple higher -derivation of a -ring . If for all and for each , then for each and for every .
Proof. The substitution of for in the definition of Jordan triple higher -derivation gives On the other hand, the substitution of for in the definition of Jordan triple higher -derivation and using our assumption that for give Now, subtracting the two relations so obtained we find that Using our notation the last relation reduces to the required result.
Now, we are ready to prove our main results.
Theorem 5. Let be a 6-torsion free semiprime -ring. Then every Jordan triple higher -derivation of is a Jordan higher -derivation of .
Proof. We intend to show that for all . In case , we get trivially for all . If , then it follows from [7, Theorem 1] that for all . Thus we assume that for all and . Thus, from Lemma 4, we see that In case is even, (8) reduces to ; by applying Lemma 2 we get . In case is odd, (8) reduces to ; by applying Lemma 3 we get . So for either of the two cases we have for each The substitution of for in relation (9) gives Substituting for in (11) we obtain Comparing (11) and (12) we get, since is 2-torsion free, that Putting for in (13) gives by the assumption that is 2-torsion free that Subtracting the relation (13) from (14) we obtain, since is 3-torsion free, that Right multiplication of (15) by and using (9) we obtain Putting for in (16) and left-multiplying by we get , for all . By the semiprimeness of it follows that for all . So (16) reduces to , for all . Again, by the semiprimeness of , we get Using (17), (15) reduces to for all . Multiplying this relation by from the right and by from the left we get for all . Again, by the semiprimeness of , we get Linearizing (17) we have Putting for in (19) we get Adding (19) and (20) we get, since is 2-torsion free, that Multiplying (21) by from the right and using (18) we get for all . By the semiprimeness of , we get for all . This completes the proof of the theorem.
Corollary 6 (see [7, Theorem 1]). Let be a 6-torsion free semiprime -ring. Then every Jordan triple -derivation of is a Jordan -derivation of .
Theorem 7. Let be a 2-torsion free -ring. Then every Jordan higher -derivation of is a Jordan triple higher -derivation of .
Proof. We have Put and using (22) we obtain Comparing the last two forms of gives Now put . Using (24) we get Also, Comparing the last two forms of and using the fact that is 2-torsion free, we obtain the required result.
Theorem 8. The notions of Jordan higher -derivation and Jordan triple higher -derivation on a 6-torsion free semiprime -ring are coincident.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author is truly indebted to Professor M. N. Daif for his constant encouragement and valuable discussions. The author also would like to express sincere gratitude to the referees for their careful reading and helpful comments. This paper is a part of the author’s Ph.D. dissertation under the supervision of Professor M. N. Daif.
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