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  • Open Access

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

Journal of the Egyptian Mathematical Society201927:4

https://doi.org/10.1186/s42787-019-0003-3

  • Received: 18 June 2018
  • Accepted: 9 September 2018
  • Published:

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 nx=0 for all xR, and of characteristic not n if nx=0, xR, then x=0. An additive subgroup L of R is called a Lie ideal of R if [u,r]L for all uL, rR. A Lie ideal L is said to be a square-closed Lie ideal of R if u2L for all uL. An R-bimodule M is a left and right R-module such that x(my)=(xm)y for all mM and x,yR. 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:RR is called a derivation if d(xy)=d(x)y+xd(y) for all x,yR. d is called a Jordan derivation in case d(x2)=d(x)x+xd(x) for all xR. Moreover, d is called a Jordan triple derivation if d(xyx)=d(x)yx+xd(y)x+xyd(x) for all x,yR. 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(x2)=f(x)x+xd(x) for all xR. 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)yx+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, yR. f is called a Jordan (θ,ϕ)–derivation if f(x2)=f(x)θ(x)+ϕ(x)f(x) for all xR. 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,yR. 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:RR is said to be a left (right) centralizer, if f(xy)=f(x)y(f(xy)=xf(y)) for all x,yR. 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 2f(xyx)=f(x)yx+xyf(x) for all x,yR, then f is a centralizer.

An additive mapping f:RR 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,yR. 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 2f(xyx)=f(x)θ(yx)+θ(xy)f(x) for all x,yR 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×RM is called a Hochschild 2–cocycle, if xα(y,z)−α(xy, z)+α(x,yz)−α(x, y)z=0 for all x,y,zR, and α is called symmetric if α(x,y)=α(y,x) for all x,yR. 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:RM 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,yR, and f is called a generalized Jordan derivation associated with α if f(x2)=f(x)x+xf(x)+α(x,x) for all xR. 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 α1:R×R(x,y)x(df)(y)M is biadditive and satisfies the 2–cocycle condition. Hence, f is a generalized derivation associated with α1.

(2) If f:RM is a left centralizer, then by f(xy)=f(x)y+xf(y)+x(−f)(y), we have a 2–cocycle α2:R×RM defined by, α2(x,y)=x(−f)(y), and hence, f is a generalized derivation associated with α2.

(3) Let f be a (θ,ϕ)−derivation. Then, the map α3: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)+α3(x,y), then f is a generalized derivation associated with α3.

(4) In general, he has mentioned the following. Let f:RM be an additive map and let α:R×RM 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.

In his work, Nakajima [8] has shown the following result. Let 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:
  1. (i)

    R is a noncommutative prime ring.

     
  2. (ii)

    There exist x,yR such that [x,y] is a nonzero divisor.

     
  3. (iii)

    R is commutative and α is symmetric.

     
Nawzad, et al. [9] have shown the following. Let 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:
  1. (i)

    R is a noncommutative semiprime ring and α is symmetric.

     
  2. (ii)

    R is commutative.

     
In [10], Rehman and Hongan have proved the following result. Let 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.
  1. (i)

    R is a prime ring and L is noncommutative.

     
  2. (ii)

    R is a prime ring, L is commutative and α is symmetric.

     
  3. (iii)

    There exist x,yR 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:RM 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,yR.

Examples (i) If f is a Jordan triple derivation, then the zero map α1 is biadditive and satisfies the 2–cocycle condition. Therefore f is a generalized Jordan triple derivation associated with α1.

(ii) If f is a generalized Jordan triple derivation associated with a Jordan triple derivation d, then α2(x,y)=x(df)(y) is biadditive and satisfies the 2–cocycle condition and we can see that f(xyx)=f(x)yx+xf(y)x+α2(x,y)x+xyf(x)+α2(xy,x). Hence f is a generalized Jordan triple derivation associated with α2.

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,zR, f(xyz+zyx)=f(x)yz+xf(y)z+α(x,y)z+zyf(z)+α(xy,z)+f(z)yx+zf(y)x+α(z,y)x+zyf(x)+α(zy,x).

Proof

Let v=f((x+z)y(x+z)), we have for all x,y,zR
$${{} \begin{aligned} 0&=v-v\\ &=f(xyx)+f(xyz+zyx)+f(zyz)- \{f(x+z)y(x+z)\\ &\,\,\,\,\, +(x+z)f(y)(x+z)+\alpha(x+z,y)(x+z)\\& \quad+(x+z)yf(x+z)\\ &\,\,\,\,\, +\alpha((x+z)y,(x+z))\}. \end{aligned}} $$
Then,
$${{} \begin{aligned} 0&=f(xyx)+f(xyz+zyx)+f(zyz)-\{f(x)yx\\&\quad+f(x)yz+f(z)yx\\ &+f(z)yz+xf(y)x+xf(y)z+zf(y)x+zf(y)z\\&\quad+\alpha(x,y)x+\alpha(x,y)z\\ &+\alpha(z,y)x+\alpha(z,y)z+xyf(x)+xyf(z)+zyf(x)\\&\quad+zyf(z)+\alpha(xy,x)\\ &+\alpha(xy,z)+\alpha(zy,x)+\alpha(zy,z)\} \mathrm{\,\,\,for \,\,\, all\,\,\,} x,y,z\in R, \end{aligned}} $$
Therefore,
$${{} \begin{aligned} f(xyz+zyx)&=f(x)yz+xf(y)z+\alpha(x,y)z+zyf(z)\\&\quad+\alpha(xy,z)\\ &\,\,\,\, +f(z)yx+zf(y)x+\alpha(z,y)x+zyf(x)\\ &\,\,\,\, +\alpha(zy,x) \mathrm{\,\,\,for\,\,\, all\,\,\,} x,y,z\in R, \end{aligned}} $$
as required. □

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)zα(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)zα(xy,z).

Lemma 2

For all x,y,z in a ring R, the following hold:
  1. (i)

    δ(x,y,z)=−δ(z,y,x), and

     
  2. (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,xR,

δ(a,b,c)xβ(a,b,c)+β(a,b,c)xδ(a,b,c)=0.

Proof

Let v=f(abcxcba+cbaxabc), then 0=vv=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
$$ {\begin{aligned} 0&=f(abc)xcba+abcf(x)cba+\alpha(abc,x)cba+abcxf(cba)\\ &+\alpha(abcx,cba)+f(cba)xabc+cbaf(x)abc+\alpha(cba,x)abc\\ &+cbaxf(abc)+\alpha(cbax,abc)-\{f(a)bcxcba+af(b)cxcba\\ &+abf(c)xcba+abcf(x)cba+ab\alpha(c,x)cba+abcxf(c)ba\\ &+ab\alpha(cx,c)ba+a\alpha(b,cxc)ba+abcxcf(b)a+a\alpha(bcxc,b)a\\ &+\alpha(a,bcxcb)a+abcxcbf(a)+\alpha(abcxcb,a)+f(c)baxabc\\ &+cf(b)axabc+cbf(a)xabc+cbaf(x)abc+cb\alpha(a,x)abc\\ &+cbaxf(a)bc+cb\alpha(ax,a)bc+c\alpha(b,axa)bc+cbaxaf(b)c\\ &+c\alpha(baxa,b)c+\alpha(c,baxab)c+cbaxabf(c)+\alpha(cbaxab,c)\}. \end{aligned}} $$
(1)
Therefore, for all a,b,c,xR
$$ {\begin{aligned} 0&=F(a,b,c)xcba+abcxF(c,b,a)\\&+ \{\alpha(abc,x)-ab\alpha(c,x)\}cba+ \{\alpha(abcx,cba)-\alpha(abcxcb,a)\}\\ &-\{ab\alpha(cx,c)ba+a\alpha(b,cxc)ba+ a\alpha(bcxc,b)a+\alpha(a,bcxcb)a\}\\ &+F(c,b,a)xabc+cbaxF(a,b,c)\\ &+\{\alpha(cba,x)-cb\alpha(a,x)\}abc+ \{\alpha(cbax,abc)-\alpha(cbaxab,c)\}\\ &-\{cb\alpha(ax,a)bc+c\alpha(b,axa)bc+ c\alpha(baxa,b)c+\alpha(c,baxab)c\} \end{aligned}} $$
(2)
Since α is a 2–cocycle map, we obtain the following relations for all a,b,c,xR:
  1. (i)

    {α((ab)c,x)−(ab)α(c,x)}cba={α(ab,cx)−α(ab,c)x}cba.

     
  2. (ii)

    α(abcx,(cb)a)−α((abcx)(cb),a)=α(abcx,cb)a−(abcx)α(cb,a).

     
  3. (iii)

    {α((cb)a,x)−(cb)α(a,x)}abc={α(cb,ax)−α(cb,a)x}abc.

     
  4. (iv)

    α(cbax,(ab)c)−α((cbax)(ab),c)=α(cbax,ab)c−(cbax)α(ab,c).

     
Substituting from (i–iv) in (2), we get for all a,b,c,xR
$$ {\begin{aligned} 0&=F(a,b,c)xcba+abcxF(c,b,a)\\ &+\{\alpha(ab,cx)-\alpha(ab,c)x\}cba+ \{\alpha(abcx,cb)a-abcx\alpha(cb,a)\}\\ &-\{ab\alpha(cx,c)ba+a\alpha(b,cxc)ba+ a\alpha(bcxc,b)a+\alpha(a,bcxcb)a\}\\ &+F(c,b,a)xabc+cbaxF(a,b,c)\\ &+\{\alpha(cb,ax)-\alpha(cb,a)x\}abc+ \{\alpha(cbax,ab)c-cbax\alpha(ab,c)\}\\ &-\{cb\alpha(ax,a)bc+c\alpha(b,axa)bc+ c\alpha(baxa,b)c+\alpha(c,baxab)c\} \end{aligned}} $$
(3)

Since α is a 2–cocycle map, we conclude for all a,b,c,xR that

(i) α(ab,cx)=aα(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)=cα(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.

Substituting from (i–iv) in (3), we obtain
$$ {\begin{aligned} 0&=\{F(a,b,c)-\alpha(ab,c)-\alpha(a,b)c\}xcba+abcx\{F(c,b,a)-\alpha(cb,a)\\ &-\alpha(c,b)a\}+\{a\alpha(b,cx)cba-ab\alpha(cx,c)ba-a\alpha(b,cxc)ba\}\\ &+\{\alpha(abcxc,b)a-a\alpha(bcxc,b)a-\alpha(a,bcxcb)a\}\\ &+\alpha(a,bcx)cba+\alpha(abcx,c)ba+\{F(c,b,a)-\alpha(cb,a)\\ &-\alpha(c,b)a\}xabc+cbax\{F(a,b,c)-\alpha(ab,c)-\alpha(a,b)c\}\\ &+\{c\alpha(b,ax)abc-cb\alpha(ax,a)bc-c\alpha(b,axa)bc\}\\ &+\{\alpha(cbaxa,b)c-c\alpha(baxa,b)c-\alpha(c,baxab)c\}\\ &+\alpha(c,bax)abc+\alpha(cbax,a)bc, \,\, \mathrm{for\,\, all}\,\ a,b,c,x\in R. \end{aligned}} $$
(4)
Again since α is a 2–cocycle map, we have
  1. (i)

    a{α(b,cx)cbα(cx,c)−α(b,(cx)c)}ba=−aα(b(cx),c)ba.

     
  2. (ii)

    {α(a(bcxc),b)−aα(bcxc,b)−α(a,(bcxc)b)}a=−α(a,bcxc)ba.

     
  3. (iii)

    c{α(b,ax)abα(ax,a)−α(b,(ax)a)}bc=−cα(b(ax),a)bc.

     
  4. (iv)

    {α(c(baxa),b)−cα(baxa,b)−α(c,(baxa)b)}c=−α(c,baxa)bc.

     
Replacing (i–iv) into (4), we get, for all a,b,c,xR
$$ {\begin{aligned} 0&=\delta(a,b,c)xcba+abcx\delta(c,b,a)-a\alpha(bcx,c)ba-\alpha(a,bcxc)ba\\ &+\alpha(a,bcx)cba+\alpha(abcx,c)ba+\delta(c,b,a)xabc+cbax\delta(a,b,c)\\ &-c\alpha(bax,a)bc-\alpha(c,baxa)bc+\alpha(c,bax)abc+\alpha(cbax,a)bc. \end{aligned}} $$
(5)
Continuing in this manner, we obtain
  1. (i)

    {−aα(bcx,c)−α(a,(bcx)c)+α(a,bcx)c+α(a(bcx),c)}ba=0.

     
  2. (ii)

    {−cα(bax,a)−α(c,(bax)a)+α(c,bax)a+α(c(bax),a)}bc=0.

     

By (5), we conclude that 0=δ(a,b,c)xcba+abcxδ(c,b,a)+δ(c,b,a)xabc+cbaxδ(a,b,c) for all a,b,c,xR. By Lemma 2, we obtain 0=δ(a,b,c)xcbaabcxδ(a,b,c)−δ(a,b,c)xabc+cbaxδ(a,b,c) for all a,b,c,xR.

Therefore, δ(a,b,c)xβ(a,b,c)+β(a,b,c)xδ(a,b,c)=0 for all a,b,c,xR. 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,xR,δ(a,b,c)xβ(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(a1,b1,c1)xβ(a2,b2,c2)=0 for all a1,b1,c1,a2,b2,c2,xR.

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,cR.

Proof

If R is commutative, then, by definition of β,β(a,b,c)=0 for all a,b,cR. Conversely, assume that β(a,b,c)=0 for all a,b,cR. 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 abccba=0 for all a,b,cQ. Replacing c by the identity of Q yields the commutativity of Q, and hence R. □

Lemma 7

Let R be a prime ring of characteristic not 2. Then δ(a,b,c)=0 for all a,b,cR, in each of the following cases:
  1. (i)

    R is noncommutative.

     
  2. ii

    There exist x,y,zR such that β(x,y,z) is a nonzero divisor in M.

     
  3. 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)rβ(x,y,z)=0 for all a,b,c,r,x,y,zR. From our assumption δ(a,b,c)r=0 for all a,b,c,rR. Thus the primeness of R gives δ(a,b,c)=0 for all a,b,cR.

(iii) From Lemma 1 we have f(abc+cba)=f(a)bc+af(b)c+α(a,b)c+abf(c)+α(ab,c)+f(c)ba+cf(b)a+α(c,b)a+cbf(a)+α(cb,a) for all a,b,cR. Since R is commutative and α is symmetric, we get 0=2{f(abc)−f(a)bcaf(b)cabf(c)}−α(a,b)cα(ab,c)−aα(b,c)−α(a,bc) for all a,b,cR. Since α is 2–cocycle we have −aα(b,c)−α(a,bc)=−α(a,b)cα(ab,c) for all a,b,cR. Therefore 0=2{f(abc)−f(a)bcaf(b)cabf(c)−α(a,b)cα(ab,c)} for all a,b,cR. Since R has characteristic not 2, then δ(a,b,c)=0 for all a,b,cR, 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,zR such that β(x,y,z) is a nonzero divisor in M.

(iii) R is commutative and α is symmetric.

Proof

Suppose that f is a generalized Jordan triple derivation associated with a Hochschild 2−cocycle α. We denote by G(a,b) and ab the elements of M defined by G(a,b)=f(ab)−f(a)baf(b), and ab=f(ab)−f(a)baf(b)−α(a,b), respectively. Thus, ab=G(a,b)−α(a,b). It is evident that ab+c=ab+ac, and (a+b)c=ac+bc. By Lemma 7, we have δ(a,b,c)=0 for all a,b,cR. Thus, for all a,b,cR
$$ f(abc)=f(a)bc+af(b)c+\alpha(a,b)c+abf(c)+\alpha(ab,c). $$
(6)
Now let v=f(abxab), then 0=vv=f((ab)x(ab))−f(a(bxa)b). By (6), we have for all a,b,xR
$${\begin{aligned} 0&=f(ab)xab+abf(x)ab+\alpha(ab,x)ab+abxf(ab)+\alpha(abx,ab)\\ &-f(a)bxab-af(b)xab-abf(x)ab-a\alpha(b,x)ab-abxf(a)b\\ &-a\alpha(bx,a)b-\alpha(a,bxa)b-abxaf(b)-\alpha(abxa,b). \end{aligned}} $$
So, for all a,b,xR
$$ {{} \begin{aligned} 0&=G(a,b)xab+abxG(a,b)+ \{\alpha(ab,x)-a\alpha(b,x)\}ab\\ &+\{\alpha(abx,ab)-\alpha(abxa,b)\} -a\alpha(bx,a)b-\alpha(a,bxa)b. \end{aligned}} $$
(7)
Since α is 2-cocycle we have for all a,b,xR that
  1. (i)

    {α(ab,x)−aα(b,x)}ab={α(a,bx)−α(a,b)x}ab, and

     
  2. (ii)

    α(abx,ab)−α((abx)a,b)=α(abx,a)b−(abx)α(a,b).

     
Substituting from (i) and (ii) in (7), we get G(a,b)xabα(a,b)xab+abxG(a,b)−abxα(a,b)+α(a,bx)ab+α(abx,a)baα(bx,a)bα(a,bxa)b=0for all a,b,xR. But α is 2–cocycle, hence {α(a,bx)a+α(abx,a)−aα(bx,a)−α(a,bxa)}b=0. Therefore abx(ab)+(ab)xab=0 for all a,b,xR. By Lemma 1.1 of Bre\(\breve {s}\)ar [3], we get
$$ a^{b}x(ab)=(ab)xa^{b}=0\,\, \mathrm{for\,\, all}\,\, a,b,x\in R. $$
(8)
Replacing a by a+c in (8) and using (8), we obtain abxcb=−cbxab for all a,b,c,xR, and then (abxcb)y(abxcb)=−abx(cbycb)xab=0 for all a,b,c,x,yR. Thus the primeness of R gives
$$ a^{b}xcb=0\,\, \mathrm{for\,\, all}\,\, a,b,c,x\in R. $$
(9)
Similarly replacing b by b+d in (9), we get
$$a^{b}xcd=0\,\, \mathrm{for\,\, all}\,\, a,b,c,d,x\in R. $$
(10)

Putting c=ab and x=dx in (10) we have abdxabd=0 for all a,b,d,xR. Again, the primeness of R yields that abd=0 for all a,b,dR, and hence ab=0 for all a,bR. Consequently, f is a generalized derivation associated with a Hochschild 2–cocycle α. □

Declarations

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.

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Both authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

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Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City, 11884, Cairo, Egypt

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