Nonlinear Jordan Triple Derivations of Triangular Algebras

In this paper, it is proved that every nonlinear Jordan triple derivation on triangular algebra is an additive derivation.


Introduction
Let  be a commutative ring with identity and  be an  -algebra.A linear map : for all , .A B ∈  Additive (linear) derivations are very important maps both in theory and applications, and were studied intensively.More generally, we say that δ is a Jordan triple derivation if A B C ∈  .If the linearity in the definition is not required, the corresponding map is said to be a nonlinear Jordan triple derivation.It should be remarked that there are several definitions of linear Jordan derivations and all of them are equivalent as long as the algebra  is 2-torsion free.We refer the reader to [1] for more details and related topics.But one can ask whether the equivalence is also true on the condition of nonlinear, and we are still unable to answer this question.
The structures of derivations, Jordan derivations and Jordan triple derivations were systematically studied.Herstein [2] proved that any Jordan derivation from a 2-torsion free prime ring into itself is a derivation, and the famous result of Brešar ([1], Theorem 4.3) states that every Jordan triple derivation from a 2-torsion free semiprime ring into itself is a derivation.For other results, see [3]- [9] and the references therein.
Let  and  be two unital algebras over a commutative ring  , and let  be a unital ( ) . Recall the algebra ( ) under the usual matrix addition and formal matrix multiplication is called a triangular algebra [10].Recently, Zhang [11] characterized that any Jordan derivation on a triangular algebra is a derivation.In this paper we present result corresponding to [11] (Theorem 2.1) for non-linear Jordan triple derivations (there is no linear or additive assumption) on an important algebra: triangular algebra.
As a notational convenience, we will adopt the traditional representations.Let us write 0 0 0 for the identity matrix of the triangular algebra  .

The Main Results
In this note, our main result is the following theorem.
Theorem 2.1.Let  and  be unital algebras over a 2-torsion free commutative ring  , and  be a unital ( ) ,   -bimodule, which is faithful as a left  -bimodule and a right  -bimodule.Let
Thus we have from the fact that ( ) 0 Proof.Firstly, we prove that ( ) (2) is proved similarly.
(3) For any 11 On the other hand, that is, ( ) On the other hand, .

P d A
Therefor combining Lemma 2.3, we have 0.   ) .Proof of Theorem 2.1.From the above lemmas, we have proved that d is an additive derivation on  .

Since ( ) ( ) ( )
for each A ∈  , by a simple calculation, we see that δ is also an additive derivation.The proof is completed.

Lemma 2 . 1 .
If δ is a nonlinear Jordan triple derivation on an upper triangular algebra  generated by

(
) On the other hand, it follows from Lemma 2.3, 2.7; we get that