Jordan Higher Derivable Mappings on Rings

Abstract

Let be a ring. We say that a family of maps is a Jordan higher derivable map (without assumption of additivity) on if (the identity map on ) and hold for all and for each . In this paper, we show that every Jordan higher derivable map on a ring under certain assumptions becomes a higher derivation. As its application, we get that every Jordan higher derivable map on Banach algebra is an additive higher derivation.

1. Introduction

Let be a ring. An additive mapping is said to be a derivation (resp., Jordan derivation) if (resp., ) holds for any . Note that, in case of -torsion free ring , is equivalent to for all . A map , (without assumption of additivity) is said to be derivable (resp., Jordan derivable) if (resp., ), holds for all . Recall that a ring is prime if implies that either or and is semiprime if implies .

Now suppose that is a ring with a nontrivial idempotent . We write . Note that need not have identity element. Put for any . Then by Peirce decomposition of , we have . Throughout this paper, the notation will denote an arbitrary element of and any element can be expressed as .

Characterizing the interrelation between the multiplicative and the additive structures of a ring is an interesting topic and has received attention of many mathematicians. It is a well-known result due to Martindale III [1] that every multiplicative bijective map from a prime ring containing a nontrivial idempotent onto an arbitrary ring is necessarily additive. Long ago, Lu [2] proved that “each derivable map on a -torsion free unital prime ring containing a nontrivial idempotent is a derivation.”

Motivated by the above result due to Lu [2], in the year 2011, Jing and Lu [3] characterized Jordan derivable maps to a larger class of rings and proved the following theorem.

Theorem 1 (see [3, Theorem 1.2]). Let be a ring containing a nontrivial idempotent and satisfying the following conditions for .(i)If for all , then .(ii)If for all , then .(iii)If for all , then .
If a mapping satisfies for all , then is additive. In addition, if is -torsion free, then is a Jordan derivation.

The concept of derivation was extended to higher derivation by Hasse and Schmidt [4] (see [5, 6] for a historical account and applications). Let be a family of additive mappings and let be the set of nonnegative integers. Following Hasse and Schmidt [4], is said to be a higher derivation (resp., Jordan higher derivation) on if (the identity map on ) and (resp., ) for all and for each . In an attempt to generalize Herstein’s result for higher derivations, Haetinger [7] proved that, on a prime ring with , every Jordan higher derivation is a higher derivation (see [6, 8] for English versions). Further many results were obtained in this direction. In the present paper we introduce the notion of higher derivable (resp., Jordan higher derivable) map on and provide a sufficient condition on a ring under which a Jordan higher derivable map becomes a higher derivation.

2. Additivity of Jordan Higher Derivable Mappings

In order to illustrate the results in the literature focused on derivable maps and higher derivation we will introduce the concept of higher derivable maps (resp., Jordan higher derivable maps) on certain classes of rings.

A family of mappings (without assumption of additivity) such that , is said to be(i)a higher derivable map if holds for all and for each ,(ii)a Jordan higher derivable map if holds for all and for each .

The main result of our paper states as follows.

Theorem 2. Let be a ring containing a nontrivial idempotent satisfying the following conditions for .)If for all , then .()If for all , then .()If for all , then .
If the family of mappings such that satisfy for all and for each , then is additive. In addition, if is -torsion free, then is a Jordan higher derivation.

Throughout the paper, we always assume that is a ring with a nontrivial idempotent and satisfies conditions , , and . We also assume that is the family of Jordan higher derivable maps ; that is,

We may see that, for , (3) reduces to for all . In view of Theorem 1, it is clear that is additive. We will use this result throughout the paper whenever needed without specific mention.

We facilitate our discussion with the following lemmas.

Lemma 3. Let be the family of Jordan higher derivable mappings . Then for each , (i),(ii),(iii),(iv).

Proof. We only prove (i). The rest of the proofs follow similarly. Since is additive, we have . Now by induction hypothesis let the result hold for all positive integer . For any , we compute On the other hand, we have Since the result holds for all , on comparing the above two equalities we obtain On using , , and , respectively, we have In order to complete the proof, we have to show that .
For any , we have Application of (3) yields that Also the above equation can be rewritten as Using hypothesis, we find that Consequently, Therefore, on application of , we obtain, . This completes the proof.

Lemma 4. Let be the family of Jordan higher derivable mappings . Then for each , (i), (ii).

Proof. (i) Using Lemma 3, we obtain (ii) Since . On using the same technique used in , we obtain the result.

Lemma 5. Let be the family of Jordan higher derivable mappings . Then for each , (i), (ii).

Proof. We prove (i); proof of (ii) follows similarly. Since is additive, we have . Now by induction hypothesis let the result hold for all positive integer . For any , we calculate in two ways.
On one hand, On the other hand, by Lemma 4, we have On comparing the above two equalities and using the hypothesis that the result holds for all , we get for all . Therefore, and . By , , and , we can deduce that We now complete the proof by showing that . Now for any , we observe that Also, Now comparing the above two expressions, we find that Therefore, we have , for all . By we can infer that . This completes the proof.

Lemma 6. Let be the family of Jordan higher derivable mappings . Then for each , (i), (ii).

Proof. We will prove (i). The proof of (ii) follows similarly. Since is additive, we have . Now by induction hypothesis let the result hold for all positive integer . For any , we have Also, On comparing the above two equalities and using the hypothesis that the result holds for all , we get This implies that Similarly, by considering and using Lemma 5 we can find that .

Lemma 7. Let be the family of Jordan higher derivable mappings . Then for each , .

Proof. Since is additive, we have . Now by induction hypothesis let the result hold for all positive integer . Consider . We have On multiplying right hand side by and using the fact that result holds good for all in the above relation, we obtain Also on using the same technique, we compute and to obtain Comparing (27), (28), and (29) we see that Now, for any , we have This further leads to Since , we see that Using and , we find that Similarly, by considering for all , we get . Consequently, .

Lemma 8. Let be the family of Jordan higher derivable mappings . Then for each , (i), (ii).

Proof. We prove (i). The proof of (ii) follows similarly. Since is additive, we have . Now by induction hypothesis let the result hold for all positive integer . For any , we have On the other hand, by Lemma 7, we also have On comparing the above two equalities, we get Hence we obtain that From and using Lemma 3, one can easily obtain that It follows that , and therefore we get .

Lemma 9. Let be the family of Jordan higher derivable mappings . Then for each , .

Proof. Since is additive, we have . Now by induction hypothesis let the result hold for all positive integer . For any , we have This gives us Now using and , the above equation reduces to Similarly one can get This completes the proof.

We are now well equipped to prove our main theorem.

Proof of Theorem 2. For any , we write and . Applying Lemmas 3–9, we have Hence we conclude that is additive.
In addition, if is -torsion free, then for any , we have Therefore, is a Jordan higher derivation.

If the underlying ring is semiprime, applying Lemma 1.5 of [3] and the fact that every Jordan higher derivation on a -torsion free semiprime ring is a higher derivation (see [6]), we have the following.

Corollary 10. Let be a -torsion free semiprime ring with a nontrivial idempotent satisfying the following conditions for . ()If for all , , then .()If for all , , then .
If the family of mappings such that satisfy for all , then is additive. Moreover, is a higher derivation.

Applying Lemma 1.6 of [3] and the fact that every Jordan higher derivation on a prime ring of characteristic different from two is a higher derivation (see the main theorem of [8]), we find the following corollary.

Corollary 11. Let be a prime ring of characteristic different from two with a nontrivial idempotent and let a family of mappings such that satisfy for all ; then is additive. Moreover, is a higher derivation.

Since every standard operator algebra is prime, we can easily have the following.

Corollary 12. Let be standard operator algebra in a Banach space whose dimension is greater than . If the family of mappings such that satisfy for all , then is additive. Moreover, is an additive higher derivation.

Conflict of Interests

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

Acknowledgment

The authors are indebted to the referees for their valuable comments and helpful suggestions.