Advances in Linear Algebra & Matrix Theory, 2014, 4, 205-209 Published Online December 2014 in SciRes. http://www.scirp.org/journal/alamt http://dx.doi.org/10.4236/alamt.2014.44018

How to cite this paper: Li, H.X. (2014) Nonlinear Jordan Triple Derivations of Triangular Algebras. Advances in Linear Alge-bra & Matrix Theory, 4, 205-209. http://dx.doi.org/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 Received 10 October 2014; revised 10 November 2014; accepted 8 December 2014

Copyright © 2014 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

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 :δ → is called a derivation if ( ) ( ) ( )AB A B A Bδ δ δ= + 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 ( ) ( ) ( ) ( ) ( ) ( ) ( )1 12 2

ABC CBA A BC A B C AB C C BA C B A CB Aδ δ δ δ δ δ δ + = + + + + +

for all , ,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 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-

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module, which is faithful as a left -bimodule, that is, for , 0 0A A A∈ = ⇒ = and a right -bimodule,

that is, for , 0 0B B B∈ = ⇒ = . Recall the algebra ( ), , : , ,0a m

Tri a b mb

= = ∈ ∈ ∈

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 0I

P = ,

0 00

QI

=

and 0

0I

II

=

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

( ), ,Tri= be the triangular algebra; if δ is a nonlinear Jordan triple derivation on , δ is an additive derivation.

Lemma 2.1. If δ is a nonlinear Jordan triple derivation on an upper triangular algebra generated by 2 ,P P= ∈ ( )P PT TPδ = − with ( )T Pδ= ∈ .

Proof. It follows from the fact 2P P= that ( ) ( ) ( ) ( )P P P P P P P Pδ δ δ δ= + + , which implies that ( ) ( ) 0.P P P Q P Qδ δ= = Thus we have from the fact that ( ) 0Q P Pδ = that ( ) ( ) ,P P P Q PT TPδ δ= = −

where ( ) .T Pδ= ∈ Now define ( ) ( ) ( )d A A AT TAδ= − − for each .A∈ Clearly, d is also a nonlinear Jordan triple deri-

vation from into itself. It follows from Lemma 2.1 that ( ) ( ) 0.d P d Q= = Lemma 2.2. ( )0 0.d = Proof. Clearly, ( ) ( ) ( ) ( )0 0 00 0 0 0 00 0 0.d d d d= + + = Lemma 2.3. ( ) ,1 2.ij ijd i j⊆ ≤ ≤ ≤ Proof. Firstly, we prove that ( ) 112 .d P ∈ It is clear that

( ) ( ) ( )12 2 2 2 ,2

d P d P P P P P P Pd P P = ⋅ ⋅ + ⋅ ⋅ =

which implies that ( ) 112 .d P ∈

Let 12 12 ,A ∈ ( ) ( ) ( ) ( )12 12 12 12 121 12 2 2 2 .2 2

d A d P A Q Q A P d P A Pd A Q = ⋅ ⋅ + ⋅ ⋅ = + Since

( ) 12 122 ,d P A ∈ we get ( )12 12 .d A ∈

Let 11 11,A ∈ ( ) ( ) ( )11 11 11 111 ,2

d A d P A P P A P Pd A P = ⋅ ⋅ + ⋅ ⋅ =

and thus ( )11 11.d A ∈

Similarly, one can check that ( )22 22 .d ⊆ Lemma 2.4. ( ) ( )2 2 0.d P d Q= = Proof. For any 12 12 ,A ∈ it follows from Lemma 2.3 that

( ) ( ) ( ) ( )12 12 12 12 121 12 2 2 2 .2 2

d A d P P A A P P d P A Pd A = ⋅ ⋅ + ⋅ ⋅ = + This implies that ( ) 122 0.Pd P A = Since

P Q is a faithful left P P -module, we have that

( )2 0.Pd P P =

It follows from ( ) 112d P ∈ , we have ( )2 0.d P = Similarly, we can get that ( )2 0.d Q = Lemma 2.5. For any 11 11 11 12 12 12 22 22 22, , , , ,A B A B A B∈ ∈ ∈ , we have (1) ( ) ( ) ( )11 12 11 12 11 12d A B d A B A d B= + , (2) ( ) ( ) ( )12 22 12 22 12 22d A B d A B A d B= + ,

(3) ( ) ( ) ( )11 11 11 11 11 11d A B d A B A d B= + , (4) ( ) ( ) ( )22 22 22 22 22 22d A B d A B A d B= + .

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Proof. (1) For any 11 11 12 12, ,A B∈ ∈ it follows from Lemma 2.3 and 2.4; we have

( ) ( )

( ) ( ) ( )

( ) ( ) ( )( ) ( )

11 12 11 12 12 11

11 12 11 12 11 12

12 11 12 11 12 11

11 12 11 12

1 2 22

1 2 2 22

2 2 2

.

d A B d P A B B A P

d P A B Pd A B A d B

d B A B d A P B A d P

d A B A d B

= ⋅ ⋅ + ⋅ ⋅

= + +

+ + + = +

(2) is proved similarly. (3) For any 11 11 11 12 12, , ,A B M∈ ∈ by Lemma 2.5 (1), we get that

( ) ( ) ( )11 11 12 11 11 12 11 11 12 .d A B M d A B M A B d M= + (1)

On the other hand,

( ) ( )( ) ( ) ( )( ) ( ) ( )

11 11 12 11 11 12 11 11 12 11 11 12

11 11 12 11 11 12 11 11 12 .

d A B M d A B M d A B M A d B M

d A B M A d B M A B d M

= = +

= + +

This and Equation (1) imply that

( ) ( ) ( )( )11 11 11 11 11 11 12 0.d A B d A B A d B M− − =

Since P Q is a faithful left P P -module and ( ) ( ) ( )11 11 11 11 11 11 11d A B d A B A d B− − ∈ , we get ( ) ( ) ( )11 11 11 11 11 11 0,d A B d A B A d B− − = that is

( ) ( ) ( )11 11 11 11 11 11 .d A B d A B A d B= +

Similarly, (4) is true for all 22 22 22,A B ∈ . Lemma 2.6. ( ) ( ) ( )11 12 11 12d A A d A d A+ = + and ( ) ( ) ( )12 22 12 22d A A d A d A+ = + . Proof. Let 11 11 12 12,A A∈ ∈ , it follows from Lemma 2.2 and 2.4, we have that

( ) ( )( ) ( )11 12 11 12 11 1210 2 2 ,2

d Q A A Q Q A A Q Qd A A Q = ⋅ + ⋅ + ⋅ + ⋅ = +

that is, ( )11 12 11 12 .d A A+ ∈ +

For any 12 12 ,X ∈ it follows from Lemma 2.5 (1), we have ( ) ( ) ( )11 12 11 12 11 12 .d A X d A X A d X= + (2)

On the other hand,

( ) ( ) ( )( )

( )( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )( ) ( )

11 12 11 12 12 12 11 12

11 12 12 11 12 12 11 12 12

12 11 12 12 11 12 12 11 12

11 12 12 11 12

1 2 22

1 2 2 22

2 2 2

.

d A X d P A A X X A A P

d P A A X Pd A A X A A d X

d X A A P X d A A P X A A d P

d A A X A d X

= ⋅ + ⋅ + ⋅ + ⋅

= + + + + +

+ + + + + + = + +

This and Equation (2) imply that ( ) ( )11 12 11 12 0.d A A d A X+ − = Since P Q is a faithful left P P -mo- dule; hence

( ) ( )11 12 11 0.P d A A d A P+ − =

Similarly, let 12 12A ∈ , for any 22 22 ,X ∈ then

( ) ( ) ( )12 22 12 22 12 22 .d A X d A X A d X= +

On the other hand,

( ) ( ) ( )( )

( ) ( )

12 22 11 12 22 22 11 12

11 12 22 12 22

1 2 22

.

d A X d P A A X X A A P

d A A X A d X

= ⋅ + ⋅ + ⋅ + ⋅

= + +

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Therefore, we get ( ) ( )11 12 12 22 0,d A A d A X+ − = that is

( ) ( ) ( ) ( )11 12 12 11 12 12 0.P d A A d A Q Q d A A d A Q+ − + + − = So

( ) ( ) ( ) ( )11 12 12 11 12 120 and 0.P d A A d A Q Q d A A d A Q+ − = + − = Therefor combining Lemma 2.3, we have

( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

11 12 11 12 11 12 11 12

11 12 11 12 11 12 11 12 0.

d A A d A d A P d A A d A d A P

P d A A d A d A Q Q d A A d A d A Q

+ − − = + − − + + − − + + − − =

that is,

( ) ( ) ( )11 12 11 12d A A d A d A+ = + . Similarly, (2) is true for all 12 12 22 22andA A∈ ∈ . Lemma 2.7. ( ) ( ) ( ) ( )11 12 22 11 12 22 .d A A A d A d A d A+ + = + + Proof. For any 11 11 12 12 22 22, ,A A A∈ ∈ ∈ ,

( ) ( ) ( )( ) ( )11 11 12 22 11 12 22 11 12 221 .2

d A d P A A A P P A A A P Pd A A A P = + + + + + = + +

( ) ( ) ( )( ) ( )12 11 12 22 11 12 22 11 12 221 2 2 .2

d A d P A A A Q Q A A A P Pd A A A Q = + + + + + = + +

( ) ( ) ( )( ) ( )22 11 12 22 11 12 22 11 12 221 .2

d A d Q A A A Q Q A A A Q Qd A A A Q = + + + + + = + +

Thus,

( ) ( ) ( )( )

( ) ( ) ( )

11 12 22 11 12 22 11 12 22

11 12 22

11 12 22

)

.

d A A A Pd A A A P Pd A A A Q

Qd A A A Q

d A d A d A

+ + = + + + + +

+ + +

= + +

Lemma 2.8. For any 12 12 12, ,A B ∈ we have ( ) ( ) ( )12 12 12 12 .d A B d A d B+ = + Proof. For any 12 12 12, ,A B ∈ from Lemma 2.3 and 2.6, we have

( ) ( ) ( ) ( ) ( )( )

( )( ) ( ) ( )( )( ) ( ) ( )( ) ( ) ( ) ( )

12 12 12 12 12 12

12 12 12 12

12 12 12 12

12 12 12 12

1 2 22

.

d A B d P P A Q B Q B P A P

Pd P A Q B P P A d Q B

d Q B P A P Q B d P A P

Pd A Q Pd B d A d B

+ = ⋅ + ⋅ + + + ⋅ + ⋅

= + + + + +

+ + + + + +

= + = +

Lemma 2.9. d is additive on 11 and 22 respectively. Proof. For any 11 11 11, ,A B ∈ by Lemma 2.5 (1), we have

( )( ) ( ) ( ) ( )11 11 12 11 11 12 11 11 12 .d A B B d A B B A B d B+ = + + + (3)

on the other hand, from Lemma 2.5 (1) and 2.8, we get that

( )( ) ( ) ( ) ( ) ( ) ( )11 11 12 11 12 11 12 11 12 11 12 11 12 11 12 .d A B B d A B B B d A B A d B d B B B d B+ = + = + + +

This and Equation (3) imply that

( ) ( ) ( )( )11 11 11 11 12 0.d A B d A d B B+ − − =

Since P Q is a faithful left P P -module and ( ) ( ) ( )11 11 11 11 11d A B d A d B+ − − ∈ , we have that ( ) ( ) ( )11 11 11 11 0,d A B d A d B+ − − = that is

( ) ( ) ( )11 11 11 11 .d A B d A d B+ = +

Similarly, we can also get the additivity of d on 22. Lemma 2.10. d is additivity. Proof. For any , ,A B∈ write 11 12 22 11 12 22, ,A A A A B B B B= + + = + + where , ,1 2.ij ij ijA B i j∈ ≤ ≤ ≤

Then Lemma 2.7-2.9 are all used in seeing the equation

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( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )

11 12 22 11 12 22

11 11 12 12 22 22

11 11 12 12 22 22

11 12 22 11 12 22

.

d A B d A A A B B B

d A B d A B d A B

d A d B d A d B d A d B

d A A A d B B B

d A d B

+ = + + + + +

= + + + + +

= + + + + +

= + + + + +

= +

Lemma 2.11. ( ) ( ) ( )d AB d A B Ad B= + for all ,A B∈ . Proof. For any , ,A B∈ let 11 12 22 11 12 22, ,A A A A B B B B= + + = + + where , ,1 2.ij ij ijA B i j∈ ≤ ≤ ≤ Now

we have that by Lemma 2.5 (1)-(4), Lemma 2.7 and 2.8

( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

11 11 11 12 12 22 22 22

11 11 11 12 12 22 22 22

11 11 11 11 11 12 11 12

12 22 12 22 22 22 22 22 .

d AB d A B A B A B A B

d A B d A B d A B d A B

d A B A d B d A B A d B

d A B A d B d A B A d B

= + + +

= + + +

= + + +

+ + + +

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

( ) ( )( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

11 12 22 11 12 22 11 12 22 11 12 22

11 11 11 11 11 12 11 12

12 22 12 22 22 22 22 22 .

d A B Ad B

d A A A B B B A A A d B B B

d A B A d B d A B A d B

d A B A d B d A B A d B

+

= + + + + + + + + +

= + + +

+ + + +

It is clear that ( ) ( ) ( )d AB d A B Ad B= + for all , .A B∈ Proof of Theorem 2.1. From the above lemmas, we have proved that d is an additive derivation on .

Since ( ) ( ) ( )d A A AT TAδ= − − for each A∈ , by a simple calculation, we see that δ 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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