Jordan {g,h}-derivations on triangular algebras

Liang Kong; Jianhua Zhang

doi:10.1515/math-2020-0044

Article Open Access

Jordan {g,h}-derivations on triangular algebras

Liang Kong and Jianhua Zhang

Published/Copyright: August 24, 2020

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 18 Issue 1

Abstract

In this article, we give a sufficient and necessary condition for every Jordan {g,h}-derivation to be a {g,h}-derivation on triangular algebras. As an application, we prove that every Jordan {g,h}-derivation on τ ( N ) is a {g,h}-derivation if and only if dim 0 + ≠ 1 or dim H − ⊥ ≠ 1 , where N is a non-trivial nest on a complex separable Hilbert space H and τ ( N ) is the associated nest algebra.

Keywords: triangular algebra; {g,h}-derivation; Jordan {g,h}-derivation

MSC 2010: 16W25; 47L35

1 Introduction

In this article, we assume that all algebras and rings are associative and 2-torsion free. Let A be an algebra over R , a commutative ring with unit. An R -linear map δ : A → A is said to be a derivation if δ ( x y ) = δ ( x ) y + x δ ( y ) for all x , y ∈ A . It is said to be a Jordan derivation if δ ( x ∘ y ) = δ ( x ) ∘ y + x ∘ δ ( y ) for all x , y ∈ A , where x ∘ y = x y + y x . Clearly, every derivation is a Jordan derivation. In general, the converse is not true (see [1]). In which algebras or rings a Jordan derivation is necessarily a derivation is a classical area of research. Herstein [2] showed that every Jordan derivation from a prime ring into itself is a derivation. Brešar [3] proved that Herstein’s result is true for semiprime rings. Zhang and Yu [4] proved that every Jordan derivation on triangular algebras is a derivation. Jordan derivations on some rings or algebras are also studied by other authors [5,6,7].

Let δ , δ 1 , δ 2 : A → A be R -linear maps. Then, δ is said to be a generalized derivation if

δ ( x y ) = δ 1 ( x ) y + x δ 2 ( y )

for all x , y ∈ A . The triples ( δ , δ 1 , δ 2 ) , where δ is a generalized derivation and δ 1 , δ 2 are R -linear maps associated with it, are called ternary derivations. Kaygorodov [8,9] described generalized derivations of semisimple Filippov algebras and ternary Mal'tsev algebras, respectively. Jiménez-Gestal and Pérez-Izquierdo [10] investigated ternary derivations of generalized Cayley-Dickson algebras. Other results related to generalized derivations and ternary derivations have also been obtained, see for example [11,12,13,14]. It should be mentioned that Brešar [15] and Nakajima [16] introduced the other two notions of generalized derivations. Komatsu and Nakajima [17] discussed the relations of the aforementioned generalized derivations.

Let f , g , h : A → A be R -linear maps. Then, f is said to be a Jordan generalized derivation with an associated map h if

f ( x ∘ y ) = f ( x ) ∘ y + x ∘ h ( y )

for all x , y ∈ A . Li and Benkovič [18] proved that every Jordan generalized derivation is a generalized derivation on triangular algebras. f is said to be a {g,h}-derivation if

f ( x y ) = g ( x ) y + x h ( y ) = h ( x ) y + x g ( y )

for all x , y ∈ A . It is clear that if f is a {g,h}-derivation, then both (f,g,h) and (f,h,g) are ternary derivations. It is said to be a Jordan {g,h}-derivation if

f ( x ∘ y ) = g ( x ) ∘ y + x ∘ h ( y )

for all x , y ∈ A . Clearly, every {g,h}-derivation is a Jordan {g,h}-derivation. Brešar [19] proved that every Jordan {g,h}-derivation of a semiprime unital algebra A is a {g,h}-derivation.

Let A and B be unital algebras over a field F whose characteristic is not two, and M be a unital ( A , B ) -bimodule, which is faithful as a left A -module and also as a right B -module. The algebra

Tri ( A , M , B ) = a m 0 b : a ∈ A , m ∈ M , b ∈ B

under the usual matrix operations is called a triangular algebra. Let U = Tri ( A , M , B ) be a triangular algebra and Z ( U ) be its center. It follows from [20] that

(1.1) Z ( U ) = a 0 0 b : a m = m b for all m ∈ M .

Let 1 A and 1 B be the identities of the algebras A and B , respectively, and let 1 be the identity of the triangular algebra U . Put

p 1 = 1 A 0 0 0 , p 2 = 1 − p 1 = 0 0 0 1 B

and

U i j = p i U p j , 1 ≤ i ≤ j ≤ 2 .

It is clear that U can be represented as U = p 1 U p 1 + p 1 U p 2 + p 2 U p 2 and so for each u ∈ U , u = u 11 + u 12 + u 22 , u i j ∈ U i j , 1 ≤ i ≤ j ≤ 2 .

Let H be a complex separable Hilbert space and B ( H ) denote the algebra of all bounded linear operators acting on H. A nest N is a totally ordered set of closed subspaces of H containing 0 and H, which is closed under the taking of arbitrary intersections and closed linear spans of its elements. For an element N ∈ N , let

N + = ⋀ { M ∈ N : M > N } and N − = ⋁ { M ∈ N : M < N } .

If N + ≠ N or N − ≠ N , then N + ⊖ N or N ⊖ N − is called an atom of N . A nest is called continuous if it has no atoms. A nest is said to be of infinite multiplicity if it has no finite dimensional atoms. It is clear that a continuous nest is of infinite multiplicity. A nest is said to be non-trivial if it contains at least one non-trivial element. The nest algebra associated with N , denoted by τ ( N ) , is the set

τ ( N ) = { T ∈ B ( H ) : T N ⊆ N for all N ∈ N } .

A nest algebra τ ( N ) is called non-trivial if N is a non-trivial nest. We refer the reader to [21] for the basic theory of nest algebras.

It is known that the triangular algebra is not semiprime. It is natural to ask if every Jordan {g,h}-derivation is a {g,h}-derivation on triangular algebras. In general, it is not true (see the proof of Theorem 3.1). The purpose of this article is to give a sufficient and necessary condition for every Jordan {g,h}-derivation to be a {g,h}-derivation on triangular algebras. As an application, we prove that every Jordan {g,h}-derivation on non-trivial nest algebras is a {g,h}-derivation if and only if dim 0 + ≠ 1 or dim H − ⊥ ≠ 1 .

2 Main result

To prove the main theorem, we need some lemmas.

Lemma 2.1

Let U be a triangular algebra and f be a Jordan {g,h}-derivation on U . Then, f ( 1 ) = g ( 1 ) + h ( 1 ) ∈ Z ( U ) .

Proof

Let f be a Jordan {g,h}-derivation, i.e.,

(2.1) f ( x ∘ y ) = g ( x ) ∘ y + x ∘ h ( y )

for all x , y ∈ U . The commutativity of the Jordan product gives

(2.2) f ( x ∘ y ) = h ( x ) ∘ y + x ∘ g ( y )

for all x , y ∈ U . Taking x = y = 1 in Eq. (2.1), we obtain f ( 1 ) = g ( 1 ) + h ( 1 ) .

For any idempotent p ∈ U , it follows from Eq. (2.1) that

(2.3) 0 = f ( p ∘ ( 1 − p ) ) = g ( p ) ∘ ( 1 − p ) + p ∘ h ( 1 − p ) = g ( p ) − g ( p ) p + g ( p ) − p g ( p ) + p h ( 1 ) − p h ( p ) + h ( 1 ) p − h ( p ) p .

Multiplying Eq. (2.3) by p from the left, we get

0 = p g ( p ) − p h ( p ) + p h ( 1 ) + p h ( 1 ) p − p g ( p ) p − p h ( p ) p .

Multiplying Eq. (2.3) by p from the right, we get

0 = g ( p ) p − h ( p ) p + h ( 1 ) p + p h ( 1 ) p − p g ( p ) p − p h ( p ) p .

Comparing the aforementioned two equalities, we have

(2.4) p g ( p ) − p h ( p ) + p h ( 1 ) = g ( p ) p − h ( p ) p + h ( 1 ) p .

Similarly, it follows from Eq. (2.2) that

(2.5) p h ( p ) − p g ( p ) + p g ( 1 ) = h ( p ) p − g ( p ) p + g ( 1 ) p .

It follows from Eqs. (2.4) and (2.5) that

p ( g ( 1 ) + h ( 1 ) ) = ( g ( 1 ) + h ( 1 ) ) p

for every idempotent p ∈ U . Hence for any u 12 ∈ U 12 , we have

( p 1 + u 12 ) ( g ( 1 ) + h ( 1 ) ) = ( g ( 1 ) + h ( 1 ) ) ( p 1 + u 12 )

and

p 1 ( g ( 1 ) + h ( 1 ) ) = ( g ( 1 ) + h ( 1 ) ) p 1 .

It follows that

p 1 ( g ( 1 ) + h ( 1 ) ) p 1 u 12 = u 12 p 2 ( g ( 1 ) + h ( 1 ) ) p 2

for all u 12 ∈ U 12 and

p 1 ( g ( 1 ) + h ( 1 ) ) p 2 = 0 .

From Eq. (1.1), we have

g ( 1 ) + h ( 1 ) = p 1 ( g ( 1 ) + h ( 1 ) ) p 1 + p 1 ( g ( 1 ) + h ( 1 ) ) p 2 + p 2 ( g ( 1 ) + h ( 1 ) ) p 2 = p 1 ( g ( 1 ) + h ( 1 ) ) p 1 + p 2 ( g ( 1 ) + h ( 1 ) ) p 2 ∈ Z ( U ) .

□

Lemma 2.2

Let U be a triangular algebra and f be a Jordan {g,h}-derivation on U . Then, [ g ( 1 ) , [ x , y ] ] = [ h ( 1 ) , [ x , y ] ] = 0 for all x , y ∈ U .

Proof

It follows from the proof of [19, Lemma 4.1] that [ h ( 1 ) − g ( 1 ) , [ x , y ] ] = 0 for all x , y ∈ U . On the other hand, we have [ h ( 1 ) + g ( 1 ) , [ x , y ] ] = 0 for all x , y ∈ U by Lemma 2.1. Hence, [ g ( 1 ) , [ x , y ] ] = [ h ( 1 ) , [ x , y ] ] = 0 for all x , y ∈ U .□

Lemma 2.3

Let U be a triangular algebra and f be a Jordan {g,h}-derivation on U . Then, there exist a derivation φ of U , ω ∈ Z ( U ) and ω 1 , ω 2 ∈ U such that

f ( x ) = φ ( x ) + ω ∘ x , g ( x ) = φ ( x ) + ω 1 ∘ x , h ( x ) = φ ( x ) + ω 2 ∘ x

for all x ∈ U .

Proof

Let ω = 1 2 f ( 1 ) , ω 1 = 1 2 g ( 1 ) , ω 2 = 1 2 h ( 1 ) . It follows from Lemma 2.1 that ω ∈ Z ( U ) . Taking y = 1 in Eq. (2.1), we obtain g ( x ) = f ( x ) − x ω 2 − ω 2 x . Taking x = 1 in Eq. (2.1), we obtain h ( y ) = f ( y ) − ω 1 y − y ω 1 . It follows from Lemma 2.2 that for all x , y ∈ U , we have

f ( x ∘ y ) = g ( x ) ∘ y + x ∘ h ( y ) = ( f ( x ) − x ω 2 − ω 2 x ) ∘ y + x ∘ ( f ( y ) − ω 1 y − y ω 1 ) = f ( x ) ∘ y + x ∘ f ( y ) − x ω y − y ω x − ω 2 x y − ω 1 y x − x y ω 1 − y x ω 2 = f ( x ) ∘ y + x ∘ f ( y ) − ω x y − ω y x − ( ω − ω 1 ) x y − ω 1 y x − x y ω 1 − y x ( ω − ω 1 ) = f ( x ) ∘ y + x ∘ f ( y ) − 2 ω ( x ∘ y ) + ω 1 [ x , y ] − [ x , y ] ω 1 = f ( x ) ∘ y + x ∘ f ( y ) − 2 ω ( x ∘ y ) + [ ω 1 , [ x , y ] ] = f ( x ) ∘ y + x ∘ f ( y ) − 2 ω ( x ∘ y ) .

Therefore,

(2.6) f ( x ∘ y ) − 2 ω ( x ∘ y ) = ( f ( x ) − 2 ω x ) ∘ y + x ∘ ( f ( y ) − 2 ω y ) .

We define a map φ : U → U by

φ ( x ) = f ( x ) − 2 ω x

for all x ∈ U . Then, φ ( x ∘ y ) = φ ( x ) ∘ y + x ∘ φ ( y ) for all x , y ∈ U by Eq. (2.6), and so φ is a Jordan derivation on U . Therefore, φ is a derivation by [4, Theorem 2.1] and

(2.7) f ( x ) = φ ( x ) + 2 ω x = φ ( x ) + ω ∘ x .

It follows from Eq. (2.7) that

f ( x y ) = f ( x ) y + x φ ( y ) = φ ( x ) y + x f ( y ) ,

which implies that f is a { f , φ } -derivation. Hence, f is a Jordan { f , φ } -derivation, i.e.,

f ( x ∘ y ) = f ( x ) ∘ y + x ∘ φ ( y ) .

On the other hand,

f ( x ∘ y ) = g ( x ) ∘ y + x ∘ h ( y ) .

Comparing the aforementioned two equalities, we have

(2.8) ( g ( x ) − f ( x ) ) ∘ y = x ∘ ( φ ( y ) − h ( y ) ) .

Taking y = 1 in Eq. (2.8), we obtain g ( x ) = f ( x ) − ω 2 ∘ x = φ ( x ) + ω 1 ∘ x . Similarly, taking x = 1 in Eq. (2.8), we obtain h ( y ) = φ ( y ) + ω 2 ∘ y .□

The main result of this article is as follows.

Theorem 2.1

Let U be a triangular algebra. Then, every Jordan {g,h}-derivation f on U is a {g,h}-derivation if and only if g ( 1 ) ∈ Z ( U ) or h ( 1 ) ∈ Z ( U ) .

Proof

We assume h ( 1 ) ∈ Z ( U ) without loss of generality by Lemma 2.1. It follows from Lemma 2.3 that

f ( x y ) = φ ( x y ) + ω ∘ ( x y ) = φ ( x ) y + x φ ( y ) + ω ∘ ( x y ) = ( g ( x ) − ω 1 ∘ x ) y + x ( h ( y ) − ω 2 ∘ y ) + ω ∘ ( x y ) = g ( x ) y + x h ( y ) + [ ω 2 , x y ] = g ( x ) y + x h ( y ) .

Similarly,

f ( x y ) = φ ( x y ) + ω ∘ ( x y ) = φ ( x ) y + x φ ( y ) + ω ∘ ( x y ) = ( h ( x ) − ω 2 ∘ x ) y + x ( g ( y ) − ω 1 ∘ y ) + ω ∘ ( x y ) = h ( x ) y + x g ( y ) + [ x y , ω 2 ] = h ( x ) y + x g ( y ) .

Therefore, f is a {g,h}-derivation.

Conversely, it follows from the definition of {g,h}-derivations that g ( 1 ) ∈ Z ( U ) and h ( 1 ) ∈ Z ( U ) .□

3 An application

As a notational convenience, if N is a closed subspace of H, let P ( N ) denote the orthogonal projection from H onto N.

Theorem 3.1

Let N be a non-trivial nest on a complex separable Hilbert space H. Then, every Jordan {g,h}-derivation on τ ( N ) is a {g,h}-derivation if and only if dim 0 + ≠ 1 or dim H − ⊥ ≠ 1 .

Proof

Let f be a Jordan {g,h}-derivation. It follows from Lemma 2.2 that

(3.1) [ h ( 1 ) , [ X , Y ] ] = 0

for all X , Y ∈ τ ( N ) . Next, we will prove h ( 1 ) ∈ Z ( τ ( N )) , and so f is a {g,h}-derivation by Theorem 2.1.

Case 1: If dim 0 + = 0 , then 0 + = 0 and so there exists a decreasing sequence N n of closed subspaces in N \ { 0 , H } such that P n = P ( N n ) converges strongly to 0. For any T ∈ B ( H ) , since P 1 T P 1 ⊥ ∈ τ ( N ) , taking X = P 1 , Y = P 1 T P 1 ⊥ in Eq. (3.1), we obtain

[ h ( 1 ) , P 1 T P 1 ⊥ ] = 0 .

Hence, P 1 h ( 1 ) P 1 + P 1 ⊥ h ( 1 ) P 1 ⊥ ∈ Z ( τ ( N ) ) . Since P n T ( P 1 − P n ) ∈ τ ( N ) , taking X = P n , Y = P n T ( P 1 − P n ) ( n ≥ 2 ) in Eq. (3.1), we obtain

0 = [ h ( 1 ) , P n T ( P 1 − P n ) ] = [ P 1 h ( 1 ) P 1 ⊥ , P n T ( P 1 − P n ) ] = − P n T ( P 1 − P n ) P 1 h ( 1 ) P 1 ⊥ .

It follows that P n B ( H ) ( P 1 − P n ) P 1 h ( 1 ) P 1 ⊥ = { 0 } and so ( P 1 − P n ) P 1 h ( 1 ) P 1 ⊥ = 0 . Letting n → ∞ , we get P 1 h ( 1 ) P 1 ⊥ = 0 . Hence, h ( 1 ) ∈ Z ( τ ( N ) ) .

Case 2: If dim H − ⊥ = 0 , then H − ⊥ = H and so there exists an increasing sequence M n of closed subspaces in N \ { 0 , H } such that Q n = P ( M n ) converges strongly to I. For any T ∈ B ( H ) , taking X = Q 1 , Y = Q 1 T Q 1 ⊥ in Eq. (3.1), we have Q 1 h ( 1 ) Q 1 + Q 1 ⊥ h ( 1 ) Q 1 ⊥ ∈ Z ( τ ( N ) ) . Since ( Q n − Q 1 ) T Q n ⊥ ∈ τ ( N ) , taking X = Q n , Y = ( Q n − Q 1 ) T Q n ⊥ ( n ≥ 2 ) in Eq. (3.1), we obtain

0 = [ h ( 1 ) , ( Q n − Q 1 ) T Q n ⊥ ] = [ Q 1 h ( 1 ) Q 1 ⊥ , ( Q n − Q 1 ) T Q n ⊥ ] = Q 1 h ( 1 ) Q 1 ⊥ ( Q n − Q 1 ) T Q n ⊥ .

It follows that Q 1 h ( 1 ) Q 1 ⊥ ( Q n − Q 1 ) B ( H ) Q n ⊥ = { 0 } and so Q 1 h ( 1 ) Q 1 ⊥ ( Q n − Q 1 ) = 0 . Letting n → ∞ , we get Q 1 h ( 1 ) Q 1 ⊥ = 0 . Hence, h ( 1 ) ∈ Z ( τ ( N )) .

Case 3: If dim 0 + > 1 , let P = P ( 0 + ) . For any T ∈ B ( H ) , taking X = P , Y = P T P ⊥ in Eq. (3.1), we have P h ( 1 ) P + P ⊥ h ( 1 ) P ⊥ ∈ Z ( τ ( N ) ) . Since P B ( H ) ⊂ τ ( N ) , we have from Eq. (3.1) that

0 = [ h ( 1 ) , [ P T P , P S P ] ] = [ P h ( 1 ) P ⊥ , [ P T P , P S P ] ] = − [ P T P , P S P ] P h ( 1 ) P ⊥

for any T , S ∈ B ( H ) . Hence,

(3.2) [ P B ( H ) P , P B ( H ) P ] P h ( 1 ) P ⊥ = { 0 } .

If dim 0 + = n , then P B ( H ) P is isomorphic to M n ( ℂ ) . Let E i j ( i , j = 1 , 2 , … , n ) be the matrix unit of order n whose ( i , j ) position is 1 and all other positions are 0. If n is odd, let

S 1 = E 11 − E n + 1 2 n + 1 2 + ∑ ( i , n − i + 1 ) ≠ n + 1 2 , n + 1 2 E i n − i + 1 .

Then, S 1 is invertible in M n ( ℂ ) and

S 1 = E 1 n + 1 2 , E n + 1 2 1 + ∑ ( i , n − i + 1 ) ≠ n + 1 2 , n + 1 2 [ E i i , E i n − i + 1 ] .

If n is even, let

S 2 = E 11 − E 22 + ∑ i = 1 n E i n − i + 1 .

Then, S 2 is invertible in M n ( ℂ ) and

S 2 = [ E 12 , E 21 ] + ∑ i = 1 n [ E i i , E i n − i + 1 ] .

Therefore, there exist A i , B i ∈ M n ( ℂ ) ( i = 1 , 2 , … , l ) such that ∑ i = 1 l [ A i , B i ] = S is invertible in M n ( ℂ ) . It follows from Eq. (3.2) that P h ( 1 ) P ⊥ = 0 and so h ( 1 ) ∈ Z ( τ ( N ) ) .

If dim 0 + = + ∞ , since [ P B ( H ) P , P B ( H ) P ] = P B ( H ) P , it follows from Eq. (3.2) that P h ( 1 ) P ⊥ = 0 and so h ( 1 ) ∈ Z ( τ ( N )) .

Case 4: If dim H − ⊥ > 1 , let Q = P ( H − ) . For any T ∈ B ( H ) , taking X = Q , Y = Q T Q ⊥ in Eq. (3.1), we have Q h ( 1 ) Q + Q ⊥ h ( 1 ) Q ⊥ ∈ Z ( τ ( N ) ) . Since B ( H ) Q ⊥ ⊂ τ ( N ) , we have from Eq. (3.1)

0 = [ h ( 1 ) , [ Q ⊥ T Q ⊥ , Q ⊥ S Q ⊥ ] ] = [ Q h ( 1 ) Q ⊥ , [ Q ⊥ T Q ⊥ , Q ⊥ S Q ⊥ ] ] = Q h ( 1 ) Q ⊥ [ Q ⊥ T Q ⊥ , Q ⊥ S Q ⊥ ]

for any T , S ∈ B ( H ) . Hence, Q h ( 1 ) Q ⊥ [ Q ⊥ B ( H ) Q ⊥ , Q ⊥ B ( H ) Q ⊥ ] = { 0 } . A similar proof to Case 3 gives h ( 1 ) ∈ Z ( τ ( N ) ) .

Conversely, if dim 0 + = 1 and dim H − ⊥ = 1 , then 0 + = H − ⊥ = ℂ . Let H 0 = H ⊖ ( ℂ ⊕ ℂ ) and so for every X ∈ τ ( N ) , X has the following form:

X = X 11 X 12 X 13 0 X 22 X 23 0 0 X 33 ,

where X 11 , X 13 , X 33 ∈ ℂ , X 12 ∈ B ( H 0 , ℂ ) , X 23 ∈ B ( ℂ , H 0 ) and X 22 ∈ B ( H 0 ) . For each X = ( X i j ) ∈ τ ( N ) , we define

f ( X ) = 2 X 11 X 12 3 2 ( X 11 − X 33 ) 0 2 X 22 X 23 0 0 2 X 33 ,

g ( X ) = X 11 0 X 11 − 2 X 33 − X 13 0 X 22 0 0 0 X 33

and

h ( X ) = X 11 0 2 X 11 − X 33 − X 13 0 X 22 0 0 0 X 33 .

It is easy to check that f is a Jordan {g,h}-derivation. However, f is not a {g,h}-derivation. If it was, then

2 0 3 2 0 0 0 0 0 0 = f 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 = g 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 + 1 0 0 0 0 0 0 0 0 h 1 0 0 0 0 0 0 0 0 = 2 0 2 0 0 0 0 0 0 ,

which is a contradiction.□

As an immediate result by Theorem 3.1, we have the following corollary.

Corollary 3.1

Let N be a non-trivial nest of infinite multiplicity. Then, every Jordan {g,h}-derivation on τ ( N ) is a {g,h}-derivation.

Acknowledgments

The authors wish to thank anonymous referees for their valuable comments and suggestions which have considerably improved the presentation of the article. This research was supported by the National Natural Science Foundation of China (11471199) and Shangluo University Key Disciplines Project, Discipline name: Mathematics.

References

[1] D. Benkovič, Jordan derivations and antiderivations on triangular matrices, Linear Algebra Appl. 397 (2005), 235–244, 10.1016/j.laa.2004.10.017.Search in Google Scholar

[2] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), no. 6, 1104–1110, 10.1090/S0002-9939-1957-0095864-2.Search in Google Scholar

[3] M. Brešar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1003–1006, 10.1090/S0002-9939-1988-0929422-1.Search in Google Scholar

[4] J. Zhang and W. Yu, Jordan derivations of triangular algebras, Linear Algebra Appl. 419 (2006), no. 1, 251–255, 10.1016/j.laa.2006.04.015.Search in Google Scholar

[5] Y. Li, L. van Wyk, and F. Wei, Jordan derivations and antiderivations of generalized matrix algebras, Oper. Matrices 7 (2013), no. 2, 399–415, 10.7153/oam-07-23.Search in Google Scholar

[6] T.-K. Lee and J.-H. Lin, Jordan derivations of prime rings with characteristic two, Linear Algebra Appl. 462 (2014), 1–15, 10.1016/j.laa.2014.08.006.Search in Google Scholar

[7] U. Sayn and F. Kuzucuoğlu, Jordan derivations of special subrings of matrix rings, Algebra Colloq. 26 (2019), no. 1, 83–92, 10.1142/S1005386719000087.Search in Google Scholar

[8] I. B. Kaygorodov, (n + 1)-ary derivations of semisimple Filippov algebras, Math. Notes 96 (2014), no. 2, 208–216, 10.1134/S0001434614070220.Search in Google Scholar

[9] I. B. Kaygorodov, On (n + 1)-ary derivations of simple n-ary Mal'tsev algebras, St. Petersburg Math. J. 25 (2014), no. 4, 575–585, 10.1090/S1061-0022-2014-01307-6.Search in Google Scholar

[10] C. Jiménez-Gestal and J. M. Pérez-Izquierdo, Ternary derivations of generalized Cayley-Dickson algebras, Comm. Algebra 31 (2003), no. 10, 5071–5094, 10.1081/AGB-120023148.Search in Google Scholar

[11] I. Kaygorodov and Y. Popov, Generalized derivations of (color) n-ary algebras, Linear Multilinear Algebra 64 (2016), no. 6, 1086–1106, 10.1080/03081087.2015.1072492.Search in Google Scholar

[12] P. D. Beites, I. Kaygorodov, and Y. Popov, Generalized derivations of multiplicative n-ary Hom-Ω color algebras, Bull. Malays. Math. Sci. Soc. 42 (2019), no. 1, 315–335, 10.1007/s40840-017-0486-8.Search in Google Scholar

[13] G. F. Leger and E. M. Luks, Generalized derivations of Lie algebras, J. Algebra 228 (2000), no. 1, 165–203, 10.1006/jabr.1999.8250.Search in Google Scholar

[14] A. I. Shestakov, Ternary derivations of Jordan superalgebras, Algebra Logic 53 (2014), no. 4, 323–348, 10.1007/s10469-014-9293-6.Search in Google Scholar

[15] M. Brešar, On the distance of the composition of two derivations to the generalized derivations, Glasg. Math. J. 33 (1991), no. 1, 89–93, 10.1017/S0017089500008077.Search in Google Scholar

[16] A. Nakajima, On categorical properties of generalized derivations, Sci. Math. 2 (1999), no. 3, 345–352.Search in Google Scholar

[17] H. Komatsu and A. Nakajima, Generalized derivations of associative algebras, Quaest. Math. 26 (2003), no. 2, 213–235, 10.2989/16073600309486055.Search in Google Scholar

[18] Y. Li and D. Benkovič, Jordan generalized derivations on triangular algebras, Linear Multilinear Algebra 59 (2011), no. 8, 841–849, 10.1080/03081087.2010.507600.Search in Google Scholar

[19] M. Brešar, Jordan {g,h}-derivations on tensor products of algebras, Linear Multilinear Algebra 64 (2016), no. 11, 2199–2207, 10.1080/03081087.2016.1145184.Search in Google Scholar

[20] W.-S. Cheung, Lie derivations of triangular algebras, Linear Multilinear Algebra 51 (2003), no. 3, 299–310, 10.1080/0308108031000096993.Search in Google Scholar

[21] K. R. Davidson, Nest Algebras, Pitman Research Notes in Mathematics Series, vol. 191, Longman, London, New York, 1988.Search in Google Scholar

Received: 2019-04-19

Revised: 2020-02-28

Accepted: 2020-05-22

Published Online: 2020-08-24

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2020-0044

Keywords for this article

triangular algebra; {g,h}-derivation; Jordan {g,h}-derivation

Creative Commons

BY 4.0