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 ![](//html.scirp.org/file/3-2230070x12.png)
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.