Advances in Linear Algebra & Matrix Theory
Vol.04 No.04(2014), Article ID:52413,4 pages
10.4236/alamt.2014.44018
Nonlinear Jordan Triple Derivations of Triangular Algebras
Hongxia Li
School of Science, Southwest University of Science and Technology, Mianyang, China
Email: 474072723@qq.com
Copyright © 2014 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
Received 10 October 2014; revised 10 November 2014; accepted 8 December 2014
ABSTRACT
In this paper, it is proved that every nonlinear Jordan triple derivation on triangular algebra is an additive derivation.
Keywords:
Nonlinear Jordan Triple Derivations, Triangular Algebras, Derivation
1. Introduction
Let be a commutative ring with identity and be an -algebra. A linear map is called a derivation if for all Additive (linear) derivations are very important maps both in theory and applications, and were studied intensively. More generally, we say that is a Jordan
triple derivation if
for all. 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 semi- prime 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 -bi- module, which is faithful as a left -bimodule, that is, for and a right -bimodule,
that is, for. 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, and for the identity matrix of the triangular algebra.
2. 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 be the triangular algebra; if is a nonlinear Jordan triple derivation on, is an additive derivation.
Lemma 2.1. If is a nonlinear Jordan triple derivation on an upper triangular algebra generated by with.
Proof. It follows from the fact that, which implies that Thus we have from the fact that that where
Now define for each Clearly, is also a nonlinear Jordan triple deri- vation from into itself. It follows from Lemma 2.1 that
Lemma 2.2.
Proof. Clearly,
Lemma 2.3.
Proof. Firstly, we prove that It is clear that
which implies that
Let Since
we get
Let and thus
Similarly, one can check that
Lemma 2.4.
Proof. For any it follows from Lemma 2.3 that
This implies that Since
is a faithful left -module, we have that
It follows from, we have Similarly, we can get that
Lemma 2.5. For any, we have
(1), (2),
(3), (4).
Proof. (1) For any it follows from Lemma 2.3 and 2.4; we have
(2) is proved similarly.
(3) For any by Lemma 2.5 (1), we get that
(1)
On the other hand,
This and Equation (1) imply that
Since is a faithful left -module and, we get that is
Similarly, (4) is true for all.
Lemma 2.6. and.
Proof. Let, it follows from Lemma 2.2 and 2.4, we have that
that is,
For any it follows from Lemma 2.5 (1), we have
(2)
On the other hand,
This and Equation (2) imply that Since is a faithful left -mo-
dule; hence
Similarly, let, for any then
On the other hand,
Therefore, we get that is
So
Therefor combining Lemma 2.3, we have
that is,
.
Similarly, (2) is true for all.
Lemma 2.7.
Proof. For any,
Thus,
Lemma 2.8. For any we have
Proof. For any from Lemma 2.3 and 2.6, we have
Lemma 2.9. is additive on and respectively.
Proof. For any by Lemma 2.5 (1), we have
(3)
on the other hand, from Lemma 2.5 (1) and 2.8, we get that
This and Equation (3) imply that
Since is a faithful left -module and, we have that that is
Similarly, we can also get the additivity of on
Lemma 2.10. is additivity.
Proof. For any write where Then Lemma 2.7-2.9 are all used in seeing the equation
Lemma 2.11. for all.
Proof. For any let where Now we have that by Lemma 2.5 (1)-(4), Lemma 2.7 and 2.8
On the other hand, it follows from Lemma 2.3, 2.7; we get that
It is clear that for all
Proof of Theorem 2.1. From the above lemmas, we have proved that is an additive derivation on. Since for each, by a simple calculation, we see that is also an additive derivation. The proof is completed.
Acknowledgements
The author would like to thank the editors and the referees for their valuable advice and kind helps.
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