# Generalized Jordan triple derivations associated with Hochschild 2–cocycles of rings

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## Abstract

In the present work, we introduce the notion of a generalized Jordan triple derivation associated with a Hochschild 2–cocycle, and we prove results which imply under some conditions that every generalized Jordan triple derivation associated with a Hochschild 2–cocycle of a prime ring with characteristic different from 2 is a generalized derivation associated with a Hochschild 2–cocycle.

## Keywords

Prime ring Derivation Generalized Jordan triple derivation Hochschild 2–cocycle## Introduction

Let *R* denote an associative ring with center *Z*(*R*). A ring *R* is said to have characteristic *n* if *n* is the least positive integer such that *n**x*=0 for all *x*∈*R*, and of characteristic not *n* if *nx*=0, *x*∈*R*, then *x*=0. An additive subgroup *L* of *R* is called a Lie ideal of *R* if [*u,r*]∈*L* for all *u*∈*L*, *r*∈*R*. A Lie ideal *L* is said to be a square-closed Lie ideal of *R* if *u*^{2}∈*L* for all *u*∈*L*. An *R*-bimodule *M* is a left and right *R*-module such that *x*(*my*)=(*xm*)*y* for all *m*∈*M* and *x,y*∈*R*. Recall that a ring *R* is called prime if *xRy*=(0) implies that either *x*=0 or *y*=0, and *R* is called semiprime if *xRx*=(0) implies *x*=0. An additive mapping *d*:*R*→*R* is called a derivation if *d*(*xy*)=*d*(*x*)*y*+*xd*(*y*) for all *x*,*y*∈*R*. *d* is called a Jordan derivation in case *d*(*x*^{2})=*d*(*x*)*x*+*xd*(*x*) for all *x*∈*R*. Moreover, *d* is called a Jordan triple derivation if *d*(*xyx*)=*d*(*x*)*y**x*+*xd*(*y*)*x*+*xyd*(*x*) for all *x,y*∈*R*. It is obvious to see that every derivation is a Jordan derivation and is a Jordan triple derivation but the converse is in general not true. A classical result of Herstein [1] asserts that any Jordan derivation of a prime ring with characteristic different from 2 is a derivation. In [2], Bre\(\breve {s}\)ar has proved Herstein’s result in the case of a semiprime ring. Also, he has shown in [3] that any Jordan triple derivation of a 2-torsion free semiprime ring is a derivation. An additive map *f* of a ring *R* is called a generalized derivation if there is a derivation *d* of *R* such that for all *x, y* in *R*, *f*(*xy*)=*f*(*x*)*y*+*xd*(*y*) and is called a generalized Jordan derivation if there is a Jordan derivation *d* such that *f*(*x*^{2})=*f*(*x*)*x*+*xd*(*x*) for all *x*∈*R*. Furthermore, *f* is said to be a generalized Jordan triple derivation if there is a Jordan triple derivation *d* of *R* such that for all *x, y* in *R*, *f*(*xyx*)=*f*(*x*)*y**x*+*xd*(*y*)*x*+*xyd*(*x*). In [4], Jing and Lu have proved in a prime ring *R* of characteristic not two that every generalized Jordan derivation of *R* is a generalized derivation, and also every generalized Jordan triple derivation on *R* is a generalized derivation.

Let *θ* and *ϕ* be endomorphisms of a ring *R*. *f* is called a (*θ*,*ϕ*)−derivation if *f*(*xy*)=*f*(*x*)*θ*(*y*)+*ϕ*(*x*)*f*(*y*) for all *x, y*∈*R*. *f* is called a Jordan (*θ*,*ϕ*)–derivation if *f*(*x*^{2})=*f*(*x*)*θ*(*x*)+*ϕ*(*x*)*f*(*x*) for all *x*∈*R*. *f* is called a Jordan triple (*θ*,*ϕ*)−derivation if *f*(*xyx*)=*f*(*x*)*θ*(*y*)*θ*(*x*)+*ϕ*(*x*)*f*(*y*)*θ*(*x*)+*ϕ*(*x*)*ϕ*(*y*)*f*(*x*) for all *x*,*y*∈*R*. In [5], Liu and Shiue have proved that every Jordan triple (*θ*,*ϕ*)−derivation on a 2-torsion free semiprime ring *R* is a (*θ*,*ϕ*)−derivation, where *θ* and *ϕ* are automorphisms. An additive mapping *f*:*R*→*R* is said to be a left (right) centralizer, if *f*(*xy*)=*f*(*x*)*y*(*f*(*xy*)=*xf*(*y*)) for all *x,y*∈*R*. *f* is called a centralizer, if *f* is both a left and right centralizer. In [6], Vukman and Kosi-Ulbl have shown that if *R* is a 2-torsion free semiprime ring and *f* is an additive mapping of *R* such that 2*f*(*xyx*)=*f*(*x*)*yx*+*xyf*(*x*) for all *x,y*∈*R*, then *f* is a centralizer.

An additive mapping *f*:*R*→*R* is said to be a left (right) *θ*−centralizer associated with a function *θ* of *R*, if *f*(*xy*)=*f*(*x*)*θ*(*y*)(*f*(*xy*)=*θ*(*x*)*f*(*y*)) for all *x,y*∈*R*. *f* is called a *θ*−centralizer, if *f* is both a left and right *θ*−centralizer. Daif, El-Sayiad, and Muthana in [7] have proved that if *R* is a 2−torsion free semiprime ring and *f* is an additive mapping of *R* such that 2*f*(*xyx*)=*f*(*x*)*θ*(*yx*)+*θ*(*xy*)*f*(*x*) for all *x,y*∈*R* with *θ*(*Z*(*R*))=*Z*(*R*), where *θ* is a nonzero surjective endomorphism on *R*, then *f* is a *θ*−centralizer.

Now let *R* be a ring and *M* be an *R*-bimodule. A biadditive map *α*:*R*×*R*→*M* is called a Hochschild 2–cocycle, if *x**α*(*y*,*z*)−*α*(*xy, z*)+*α*(*x,yz*)−*α*(*x, y*)*z*=0 for all *x,y,z*∈*R*, and *α* is called symmetric if *α*(*x,y*)=*α*(*y,x*) for all *x*,*y*∈*R*. Nakajima [8] has introduced a new type of generalized derivations and generalized Jordan derivations associated with Hochschild 2–cocycles in the following way. An additive map *f*:*R*→*M* is called a generalized derivation associated with a Hochschild 2–cocycle *α* if *f*(*xy*)=*f*(*x*)*y*+*xf*(*y*)+*α*(*x*,*y*) for all *x,y*∈*R*, and *f* is called a generalized Jordan derivation associated with *α* if *f*(*x*^{2})=*f*(*x*)*x*+*xf*(*x*)+*α*(*x,x*) for all *x*∈*R*. If *α*=0, then *f* means the usual derivation and Jordan derivation. He has given the following examples:

(1) If *f* is a generalized derivation associated with a derivation *d*, then the map *α*_{1}:*R*×*R*∋(*x*,*y*)↦*x*(*d*−*f*)(*y*)∈*M* is biadditive and satisfies the 2–cocycle condition. Hence, *f* is a generalized derivation associated with *α*_{1}.

(2) If *f*:*R*→*M* is a left centralizer, then by *f*(*xy*)=*f*(*x*)*y*+*xf*(*y*)+*x*(−*f*)(*y*), we have a 2–cocycle *α*_{2}:*R*×*R*→*M* defined by, *α*_{2}(*x,y*)=*x*(−*f*)(*y*), and hence, *f* is a generalized derivation associated with *α*_{2}.

(3) Let *f* be a (*θ*,*ϕ*)−derivation. Then, the map *α*_{3}:*R*×*R*∋(*x*,*y*)↦*f*(*x*)(*θ*(*y*)−*y*)+(*ϕ*(*x*)−*x*)*f*(*y*)∈*M*, is biadditive and satisfies the 2–cocycle condition. Since *f*(*xy*)=*f*(*x*)*y*+*xf*(*y*)+*α*_{3}(*x,y*), then *f* is a generalized derivation associated with *α*_{3}.

(4) In general, he has mentioned the following. Let *f*:*R*→*M* be an additive map and let *α*:*R*×*R*→*M* be a biadditive map. If *f*(*xy*)=*f*(*x*)*y*+*xf*(*y*)+*α*(*x,y*) holds, then by the associativity *f*((*xy*)*z*)=*f*(*x*(*yz*)), *α* satisfies the 2–cocycle condition. Thus *f* is a generalized derivation associated with *α*.

*R*be a 2-torsion free ring. Then, every generalized Jordan derivation associated with a Hochschild 2–cocycle

*α*is a generalized derivation associated with

*α*in each of the following cases:

- (i)
*R*is a noncommutative prime ring. - (ii)
There exist

*x,y*∈*R*such that [*x,y*] is a nonzero divisor. - (iii)
*R*is commutative and*α*is symmetric.

*R*be a 2-torsion free ring. Then, every generalized Jordan derivation associated with a Hochschild 2–cocycle

*α*is a generalized derivation associated with

*α*in each of the following cases:

- (i)
*R*is a noncommutative semiprime ring and*α*is symmetric. - (ii)
*R*is commutative.

*R*be a 2-torsion free ring and

*L*a square-closed Lie ideal of

*R*. Then, every generalized Jordan derivation associated with a Hochschild 2–cocycle

*α*is a generalized derivation associated with

*α*in each of the following cases.

- (i)
*R*is a prime ring and*L*is noncommutative. - (ii)
*R*is a prime ring,*L*is commutative and*α*is symmetric. - (iii)
There exist

*x,y*∈*R*such that [*x,y*] is a nonzero divisor in*L*.

In the present article, we introduce the notion of generalized Jordan triple derivations associated with Hochschild 2–cocycles in the following way. Let *R* be a ring and let *M* be an *R*-bimodule. An additive map *f*:*R*→*M* is called a generalized Jordan triple derivation associated with a Hochschild 2–cocycle *α* if *f*(*xyx*)=*f*(*x*)*yx*+*xf*(*y*)*x*+*α*(*x,y*)*x*+*xyf*(*x*)+*α*(*xy,x*) for all *x*,*y*∈*R*.

**Examples** (i) If *f* is a Jordan triple derivation, then the zero map *α*_{1} is biadditive and satisfies the 2–cocycle condition. Therefore *f* is a generalized Jordan triple derivation associated with *α*_{1}.

(ii) If *f* is a generalized Jordan triple derivation associated with a Jordan triple derivation *d*, then *α*_{2}(*x,y*)=*x*(*d*−*f*)(*y*) is biadditive and satisfies the 2–cocycle condition and we can see that *f*(*xyx*)=*f*(*x*)*yx*+*xf*(*y*)*x*+*α*_{2}(*x,y*)*x*+*xyf*(*x*)+*α*_{2}(*xy,x*). Hence *f* is a generalized Jordan triple derivation associated with *α*_{2}.

Our aim in this work is to show that every generalized Jordan triple derivation associated with a Hochschild 2–cocycle *α* from a prime ring *R* with characteristic different from 2 to an *R*-bimodule *M* is a generalized derivation associated with *α*.

## Preliminary results

The proof of our result is based on the following series of auxiliary lemmas.

### **Lemma 1**

Let *f* be a generalized Jordan triple derivation from a ring *R* to an *R*- bimodule *M* associated with a Hochschild 2–cocycle map *α* from *R*×*R* into *M*. Then for all *x*,*y*,*z*∈*R*, *f*(*xyz*+*zyx*)=*f*(*x*)*yz*+*xf*(*y*)*z*+*α*(*x,y*)*z*+*zyf*(*z*)+*α*(*xy,z*)+*f*(*z*)*y**x*+*zf*(*y*)*x*+*α*(*z*,*y*)*x*+*zyf*(*x*)+*α*(*zy,x*).

### *Proof*

*v*=

*f*((

*x*+

*z*)

*y*(

*x*+

*z*)), we have for all

*x*,

*y*,

*z*∈

*R*

For a generalized Jordan triple derivation *f* from a ring *R* to an *R*-bimodule *M* associated with a Hochschild 2−cocycle *α*, we denote by *δ*, *F* and *β* the maps from *R*×*R*×*R* into *M* defined by *δ*(*x,y,z*)=*f*(*xyz*)−*f*(*x*)*yz-xf*(*y*)*z*−*α*(*x,y*)*z-xyf*(*z*)−*α*(*xy,z*), *F*(*x,y,z*)=*f*(*xyz*)−*f*(*x*)*yz-xf*(*y*)*z-xyf*(*z*) and *β*(*x,y,z*)=*xyz-zyx*, respectively. Thus, *δ*(*x,y,z*)=*F*(*x,y,z*)−*α*(*x,y*)*z*−*α*(*xy,z*).

### **Lemma 2**

*x,y,z*in a ring

*R*, the following hold:

- (i)
*δ*(*x,y,z*)=−*δ*(*z,y,x*), and - (ii)
*δ*(*x,y,z*) and*β*(*x,y,z*) are tri-additive.

### *Proof*

(i) Follows easily from Lemma 1.

(ii) Replace *x* by *a*+*b* in the definition of *δ*, then (ii) is easily seen. □

### **Lemma 3**

For any ring *R* and any *a,b,c,x*∈*R*,

*δ*(*a,b,c*)*x**β*(*a,b,c*)+*β*(*a,b,c*)*x**δ*(*a*,*b*,*c*)=0.

### *Proof*

*v*=

*f*(

*abcxcba+cbaxabc*), then 0=

*v*−

*v*=

*f*((

*abc*)

*x*(

*cba*)+(

*cba*)

*x*(

*abc*))−

*f*(

*a*(

*bcxcb*)

*a*+

*c*(

*baxab*)

*c*). By the definition of the generalized Jordan triple derivation

*f*associated with a Hochschild 2-cocycle

*α*and by Lemma 1, we get

*a,b,c,x*∈

*R*

*α*is a 2–cocycle map, we obtain the following relations for all

*a,b,c,x*∈

*R*:

- (i)
{

*α*((*a**b*)*c*,*x*)−(*a**b*)*α*(*c*,*x*)}*c**b**a*={*α*(*a**b*,*c**x*)−*α*(*a**b*,*c*)*x*}*c**b**a*. - (ii)
*α*(*a**b**c**x*,(*c**b*)*a*)−*α*((*a**b**c**x*)(*c**b*),*a*)=*α*(*a**b**c**x*,*c**b*)*a*−(*a**b**c**x*)*α*(*c**b*,*a*). - (iii)
{

*α*((*c**b*)*a*,*x*)−(*c**b*)*α*(*a*,*x*)}*a**b**c*={*α*(*c**b*,*a**x*)−*α*(*c**b*,*a*)*x*}*a**b**c*. - (iv)
*α*(*c**b**a**x*,(*a**b*)*c*)−*α*((*c**b**a**x*)(*a**b*),*c*)=*α*(*c**b**a**x*,*a**b*)*c*−(*c**b**a**x*)*α*(*a**b*,*c*).

*a,b,c,x*∈

*R*

Since *α* is a 2–cocycle map, we conclude for all *a,b,c,x*∈*R* that

(i) *α*(*ab,cx*)=*a**α*(*b,cx*)+*α*(*a,b*(*cx*))−*α*(*a,b*)(*cx*).

(ii) *α*(*abcx,cb*)*a*={−(*abcx*)*α*(*c,b*)+*α*((*abcx*)*c,b*)+*α*(*abcx,c*)*b*}*a*.

(iii) *α*(*cb,ax*)=*c**α*(*b,ax*)+*α*(*c,b*(*ax*))−*α*(*c,b*)(*ax*).

(iv) *α*(*cbax,ab*)*c*={−(*cbax*)*α*(*a,b*)+*α*((*cbax*)*a,b*)+*α*(*cbax,a*)*b*}*c*.

*α*is a 2–cocycle map, we have

- (i)
*a*{*α*(*b*,*c**x*)*c*−*b**α*(*c**x*,*c*)−*α*(*b*,(*c**x*)*c*)}*b**a*=−*a**α*(*b*(*c**x*),*c*)*b**a*. - (ii)
{

*α*(*a*(*b**c**x**c*),*b*)−*a**α*(*b**c**x**c*,*b*)−*α*(*a*,(*b**c**x**c*)*b*)}*a*=−*α*(*a*,*b**c**x**c*)*b**a*. - (iii)
*c*{*α*(*b*,*a**x*)*a*−*b**α*(*a**x*,*a*)−*α*(*b*,(*a**x*)*a*)}*b**c*=−*c**α*(*b*(*a**x*),*a*)*b**c*. - (iv)
{

*α*(*c*(*b**a**x**a*),*b*)−*c**α*(*b**a**x**a*,*b*)−*α*(*c*,(*b**a**x**a*)*b*)}*c*=−*α*(*c*,*b**a**x**a*)*b**c*.

*a*,

*b*,

*c*,

*x*∈

*R*

- (i)
{−

*a**α*(*b**c**x*,*c*)−*α*(*a*,(*b**c**x*)*c*)+*α*(*a*,*b**c**x*)*c*+*α*(*a*(*b**c**x*),*c*)}*b**a*=0. - (ii)
{−

*c**α*(*b**a**x*,*a*)−*α*(*c*,(*b**a**x*)*a*)+*α*(*c*,*b**a**x*)*a*+*α*(*c*(*b**a**x*),*a*)}*b**c*=0.

By (5), we conclude that 0=*δ*(*a*,*b*,*c*)*x**c**b**a*+*a**b**c**x**δ*(*c*,*b*,*a*)+*δ*(*c*,*b*,*a*)*x**a**b**c*+*c**b**a**x**δ*(*a*,*b*,*c*) for all *a*,*b*,*c*,*x*∈*R*. By Lemma 2, we obtain 0=*δ*(*a*,*b*,*c*)*x**c**b**a*−*a**b**c**x**δ*(*a*,*b*,*c*)−*δ*(*a*,*b*,*c*)*x**a**b**c*+*c**b**a**x**δ*(*a*,*b*,*c*) for all *a*,*b*,*c*,*x*∈*R*.

Therefore, *δ*(*a*,*b*,*c*)*x**β*(*a*,*b*,*c*)+*β*(*a*,*b*,*c*)*x**δ*(*a*,*b*,*c*)=0 for all *a*,*b*,*c*,*x*∈*R*. This finishes the proof of the lemma. □

### **Lemma 4**

If *R* is a prime ring of characteristic not 2, then for all *a*,*b*,*c*,*x*∈*R*,*δ*(*a*,*b*,*c*)*x**β*(*a*,*b*,*c*)=0,.

### *Proof*

By Lemma 3 and Lemma 1.1 of Bre\(\breve {s}\)ar [3], we get the proof. □

### **Lemma 5**

If *R* is a prime ring of characteristic not 2, then

*δ*(*a*_{1},*b*_{1},*c*_{1})*x**β*(*a*_{2},*b*_{2},*c*_{2})=0 for all *a*_{1},*b*_{1},*c*_{1},*a*_{2},*b*_{2},*c*_{2},*x*∈*R*.

### *Proof*

From Lemma 2(ii), Lemma 4, and Lemma 1.2 of Bre\(\breve {s}\)ar [3], we get the proof. □

### **Lemma 6**

Let *R* be a prime ring. Then, *R* is commutative iff *β*(*a*,*b*,*c*)=0 for all *a*,*b*,*c*∈*R*.

### *Proof*

If *R* is commutative, then, by definition of *β*,*β*(*a*,*b*,*c*)=0 for all *a*,*b*,*c*∈*R*. Conversely, assume that *β*(*a*,*b*,*c*)=0 for all *a*,*b*,*c*∈*R*. Let *Q* be the Martindale right ring of quotients of *R* defined by Martindale [11]. Then *Q* is a prime ring with identity that contains the ring *R*. By Chuang [12], *Q* satisfies the same generalized polynomial identities as *R*. In particular *a**b**c*−*c**b**a*=0 for all *a*,*b*,*c*∈*Q*. Replacing *c* by the identity of *Q* yields the commutativity of *Q*, and hence *R*. □

### **Lemma 7**

*R*be a prime ring of characteristic not 2. Then

*δ*(

*a*,

*b*,

*c*)=0 for all

*a*,

*b*,

*c*∈

*R*, in each of the following cases:

- (i)
*R*is noncommutative. - ii
There exist

*x*,*y*,*z*∈*R*such that*β*(*x*,*y*,*z*) is a nonzero divisor in*M*. - iii
*R*is commutative and*α*is symmetric.

### *Proof*

(i) By Lemmas 5 and 6, we get our requirement.

(ii) By Lemma 5, we have *δ*(*a*,*b*,*c*)*r**β*(*x*,*y*,*z*)=0 for all *a*,*b*,*c*,*r*,*x*,*y*,*z*∈*R*. From our assumption *δ*(*a*,*b*,*c*)*r*=0 for all *a*,*b*,*c*,*r*∈*R*. Thus the primeness of *R* gives *δ*(*a*,*b*,*c*)=0 for all *a*,*b*,*c*∈*R*.

(iii) From Lemma 1 we have *f*(*a**b**c*+*c**b**a*)=*f*(*a*)*b**c*+*a**f*(*b*)*c*+*α*(*a*,*b*)*c*+*a**b**f*(*c*)+*α*(*a**b*,*c*)+*f*(*c*)*b**a*+*c**f*(*b*)*a*+*α*(*c*,*b*)*a*+*c**b**f*(*a*)+*α*(*c**b*,*a*) for all *a*,*b*,*c*∈*R*. Since *R* is commutative and *α* is symmetric, we get 0=2{*f*(*a**b**c*)−*f*(*a*)*b**c*−*a**f*(*b*)*c*−*a**b**f*(*c*)}−*α*(*a*,*b*)*c*−*α*(*a**b*,*c*)−*a**α*(*b*,*c*)−*α*(*a*,*b**c*) for all *a*,*b*,*c*∈*R*. Since *α* is 2–cocycle we have −*a**α*(*b*,*c*)−*α*(*a*,*b**c*)=−*α*(*a*,*b*)*c*−*α*(*a**b*,*c*) for all *a*,*b*,*c*∈*R*. Therefore 0=2{*f*(*a**b**c*)−*f*(*a*)*b**c*−*a**f*(*b*)*c*−*a**b**f*(*c*)−*α*(*a*,*b*)*c*−*α*(*a**b*,*c*)} for all *a*,*b*,*c*∈*R*. Since *R* has characteristic not 2, then *δ*(*a*,*b*,*c*)=0 for all *a*,*b*,*c*∈*R*, as required. □

## Main result

### **Theorem 1**

Let *R* be a prime ring of characteristic not 2. Then every generalized Jordan triple derivation associated with a Hochschild 2–cocycle *α* is a generalized derivation associated with *α* in each of the following cases.

(i) *R* is noncommutative.

(ii) There exist *x*,*y*,*z*∈*R* such that *β*(*x*,*y*,*z*) is a nonzero divisor in *M*.

(iii) *R* is commutative and *α* is symmetric.

### *Proof*

*f*is a generalized Jordan triple derivation associated with a Hochschild 2−cocycle

*α*. We denote by

*G*(

*a*,

*b*) and

*a*

^{b}the elements of

*M*defined by

*G*(

*a*,

*b*)=

*f*(

*a*

*b*)−

*f*(

*a*)

*b*−

*a*

*f*(

*b*), and

*a*

^{b}=

*f*(

*a*

*b*)−

*f*(

*a*)

*b*−

*a*

*f*(

*b*)−

*α*(

*a*,

*b*), respectively. Thus,

*a*

^{b}=

*G*(

*a*,

*b*)−

*α*(

*a*,

*b*). It is evident that

*a*

^{b+c}=

*a*

^{b}+

*a*

^{c}, and (

*a*+

*b*)

^{c}=

*a*

^{c}+

*b*

^{c}. By Lemma 7, we have

*δ*(

*a*,

*b*,

*c*)=0 for all

*a*,

*b*,

*c*∈

*R*. Thus, for all

*a*,

*b*,

*c*∈

*R*

*v*=

*f*(

*a*

*b*

*x*

*a*

*b*), then 0=

*v*−

*v*=

*f*((

*a*

*b*)

*x*(

*a*

*b*))−

*f*(

*a*(

*b*

*x*

*a*)

*b*). By (6), we have for all

*a*,

*b*,

*x*∈

*R*

*a*,

*b*,

*x*∈

*R*

*α*is 2-cocycle we have for all

*a*,

*b*,

*x*∈

*R*that

- (i)
{

*α*(*a**b*,*x*)−*a**α*(*b*,*x*)}*a**b*={*α*(*a*,*b**x*)−*α*(*a*,*b*)*x*}*a**b*, and - (ii)
*α*(*a**b**x*,*a**b*)−*α*((*a**b**x*)*a*,*b*)=*α*(*a**b**x*,*a*)*b*−(*a**b**x*)*α*(*a*,*b*).

*G*(

*a*,

*b*)

*x*

*a*

*b*−

*α*(

*a*,

*b*)

*x*

*a*

*b*+

*a*

*b*

*x*

*G*(

*a*,

*b*)−

*a*

*b*

*x*

*α*(

*a*,

*b*)+

*α*(

*a*,

*b*

*x*)

*a*

*b*+

*α*(

*a*

*b*

*x*,

*a*)

*b*−

*a*

*α*(

*b*

*x*,

*a*)

*b*−

*α*(

*a*,

*b*

*x*

*a*)

*b*=0for all

*a*,

*b*,

*x*∈

*R*. But

*α*is 2–cocycle, hence {

*α*(

*a*,

*b*

*x*)

*a*+

*α*(

*a*

*b*

*x*,

*a*)−

*a*

*α*(

*b*

*x*,

*a*)−

*α*(

*a*,

*b*

*x*

*a*)}

*b*=0. Therefore

*a*

^{b}

*x*(

*a*

*b*)+(

*a*

*b*)

*x*

*a*

^{b}=0 for all

*a*,

*b*,

*x*∈

*R*. By Lemma 1.1 of Bre\(\breve {s}\)ar [3], we get

*a*by

*a*+

*c*in (8) and using (8), we obtain

*a*

^{b}

*x*

*c*

*b*=−

*c*

^{b}

*x*

*a*

*b*for all

*a*,

*b*,

*c*,

*x*∈

*R*, and then (

*a*

^{b}

*x*

*c*

*b*)

*y*(

*a*

^{b}

*x*

*c*

*b*)=−

*a*

^{b}

*x*(

*c*

*b*

*y*

*c*

^{b})

*x*

*a*

*b*=0 for all

*a*,

*b*,

*c*,

*x*,

*y*∈

*R*. Thus the primeness of

*R*gives

*b*by

*b*+

*d*in (9), we get

Putting *c*=*a*^{b} and *x*=*d**x* in (10) we have *a*^{b}*d**x**a*^{b}*d*=0 for all *a*,*b*,*d*,*x*∈*R*. Again, the primeness of *R* yields that *a*^{b}*d*=0 for all *a*,*b*,*d*∈*R*, and hence *a*^{b}=0 for all *a*,*b*∈*R*. Consequently, *f* is a generalized derivation associated with a Hochschild 2–cocycle *α*. □

## Notes

### Acknowledgements

The authors are very grateful to Prof. M. N. Daif for his helpful comments and suggestions. This paper is a part of the second author’s Ph.D. dissertation under the supervision of Prof M. N. Daif.

### Funding

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### Availability of data and materials

Not applicable.

### Authors’ contributions

Both authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

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