# On Generalized Jordan Triple(σ,τ)-Higher Derivations in Prime Rings

Authors: Mohammad Ashraf; Almas Khan
Journal: ISRN Algebra
 (2014)

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## Abstract

LetR be a ring and let U be a Lie ideal of R. Suppose that σ,τ are endomorphisms of R, and ℕ is the set of all nonnegative integers. A family F={fn}n∈ℕ of mappings fn:R→R is said to be a generalized (σ,τ)-higher derivation (resp., generalized Jordan triple (σ,τ)-higher derivation) of R if there exists a (σ,τ)-higher derivation D={dn}n∈ℕ of R such that f0=IR, the identity map on R, fn(a+b)=fn(a)+fn(b), and fn(ab)=∑i+j=nfi(σn-i(a))dj(τn-j(b)) (resp., fn(aba)=∑i+j+k=nfi(σn-i(a))dj(σkτi(b))dk(τn-k(a))) hold for all a,b∈R and for every n∈ℕ. If the above conditions hold for all a,b∈U, then F is said to be a generalized (σ,τ)-higher derivation (resp., generalized Jordan triple (σ,τ)-higher derivation) of U into R. In the present paper it is shown that if U is a noncentral square closed Lie ideal of a prime ring R of characteristic different from two, then every generalized Jordan triple (σ,τ)-higher derivation of U into R is a generalized (σ,τ)-higher derivation of U into R.


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## Body

## 1. Introduction

LetR be an associative ring with center Z(R) (may be without identity element). For any x,y,z∈R, [x,y]=xy-yx will denote the usual Lie product and the element xyz-zyx will be denoted by [x,y,z]. An additive subgroup U of R is said to be a Lie ideal of R if [U,R]⊆U. A Lie ideal U of R is said to be square closed Lie ideal of R if u2∈U for all u∈U. An additive mapping d:R→R is said to be a derivation on R if d(ab)=d(a)b+ad(b) for all a,b∈R. Also, an additive mapping d:R→R is called a Jordan triple derivation if d(aba)=d(a)ba+ad(b)a+abd(a) for all a,b∈R. Herstein [1, Lemma 3.5] showed that on a 2-torsion free ring every derivation is Jordan triple derivation, but the converse need not be true in general, whereas Brešar [2] proved that on a 2-torsion free semiprime ring every Jordan triple derivation is a derivation.The concept of derivations was further extended to(σ,τ)-derivation. Let σ,τ be the endomorphisms of R. An additive mapping d:R→R is said to be a (σ,τ)-derivation of R if d(xy)=d(x)τ(y)+σ(x)d(y) holds for all x,y∈R. Moreover, an additive mapping d:R→R is said to be a Jordan triple (σ,τ)-derivation of R if d(xyx)=d(x)τ(yx)+σ(a)d(b)τ(a)+σ(ab)d(a) holds for all x,y∈R. Brešar [3] introduced the notion of generalized derivation as follows: an additive mapping f:R→R is said to be a generalized derivation on R if there exists a derivation d on R such that f(xy)=f(x)y+xd(y) for all x,y∈R. An additive mapping f:R→R is said to be a generalized (σ,τ)-derivation on R if there exists a (σ,τ)-derivation d on R such that f(xy)=f(x)τ(y)+σ(x)d(y) for all x,y∈R (for reference see [4]). Correspondingly, the concept of generalized Jordan triple derivations was defined by Wu and Lu [5]. An additive mapping f:R→R is said to be a generalized Jordan triple derivation on R if there exists a Jordan triple derivation d on R such that f(aba)=f(a)ba+ad(b)a+abd(a) for all a,b∈R. They also showed that every generalized Jordan triple derivation on a 2-torsion free prime ring is a generalized derivation. Further, Liu and Shiue [6] extended this result for generalized Jordan triple (σ,τ)-derivation. An additive mapping f:R→R is said to be a generalized Jordan triple (σ,τ)-derivation on R if there exists a Jordan triple (σ,τ)-derivation d on R such that f(aba)=f(a)τ(ba)+σ(a)d(b)τ(a)+σ(ab)d(a) for all a,b∈R. Liu and Shiue proved that on a 2-torsion free semiprime ring every generalized Jordan triple (σ,τ)-derivation is a generalized (σ,τ)-derivation.Various results proved for derivations and generalized derivations were shown to be true in case of higher derivations which can be found in [7–10], and so forth. Let ℕ be the set of all nonnegative integers. Following Hasse and Schmidt [11], a family of additive mappings D={dn}n∈ℕ on R is said to be a higher derivation on R if d0=IR  (the identity map on R) and dn(ab)=∑i+j=ndi(a)dj(b) holds for all a,b∈R and for each n∈ℕ.The concept of higher derivation was extended to(σ,τ)-higher derivation by the authors together with Haetinger [7]. Let D={dn}n∈ℕ be a family of maps dn:R→R. Then D is said to be a (σ,τ)-higher derivation on R  if  d0=IR, dn(a+b)=dn(a)+dn(b) and dn(ab)=∑i+j=ndi(σn-i(a))dj(τn-j(b))  hold for all a,b∈R and for each n∈ℕ. If U is a Lie ideal of R, then D is said to be a (σ,τ)-higher derivation of U into R if the above conditions hold for all a,b∈U. Further, in [8] the authors introduced the notion of generalized (σ,τ)-higher derivation in rings. A family F={fn}n∈ℕ of mappings fn:R→R is said to be a generalized (σ,τ)-higher derivation of R if there exists a (σ,τ)-higher derivation D={dn}n∈ℕ of R such that f0=IR, fn(a+b)=fn(a)+fn(b) and fn(ab)=∑i+j=nfi(σn-i(a))dj(τn-j(b)) for all a,b∈R and for each n∈ℕ.Motivated by the concepts of Jordan triple derivation, generalized derivation, and(σ,τ)-higher derivation, we introduce generalized Jordan triple (σ,τ)-higher derivation as follows: a family F={fn}n∈ℕ of mappings fn:R→R is said to be a generalized Jordan triple (σ,τ)-higher derivation of R if there exists a (σ,τ)-higher derivation D={dn}n∈ℕ of R such that fn(a+b)=fn(a)+fn(b) and fn(aba)=∑i+j+k=nfi(σn-i(a))dj(σkτi(b))dk(τn-k(a))  hold for all  a,b∈R and every n∈ℕ. Further, if D is a (σ,τ)-higher derivation of U into R and the above conditions hold for all a,b∈U, then F is said to be a generalized Jordan triple (σ,τ)-higher derivation of U into R.It can be easily seen that, on a 2-torsion free ringR, every generalized (σ,τ)-higher derivation of a square closed Lie ideal U into R is a generalized Jordan triple (σ,τ)-higher derivation of U into R. In fact, if F={fn}n∈ℕ is generalized (σ,τ)-higher derivation of a square closed Lie ideal U into R, then fn(uv)=∑i+j=nfi(σn-i(u))dj(τn-j(v)) for all u,v∈U and for each n∈ℕ. Replacing v with 2vu and using the fact that R is 2-torsion free, we obtain fn(uvu)=∑i+j=nfi(σn-i(u))dj(σkτi(v))dk(τn-k(u)) for all u,v∈U and for each  n∈ℕ; that is, F={fn}n∈ℕ is generalized Jordan triple (σ,τ)-higher derivation of  U into R but the converse need not be true in general.In the present paper, our objective is to find the conditions onR under which every generalized Jordan triple (σ,τ)-higher derivation of  U into R is a generalized (σ,τ)-higher derivation of  U into R. In fact our result generalizes the main theorem obtained in [4, 7–9, 12–14].


## 2. Main Results

Throughout,σ,τ will denote the endomorphisms of R such that τ is one-one and onto and στ=τσ. Note that if a Lie ideal  U of R is square closed, then for all u,v∈U;uv+vu=(u+v)2-u2-v2∈U and since uv-vu∈U; yields 2uv∈U. Let F={fn}n∈ℕ be a generalized Jordan triple (σ,τ)-higher derivation of  U into R with associated (σ,τ)-higher derivation D={dn}n∈ℕ of U into R. For every fixed n∈ℕ and each u,v,w∈U, we denote by Φn(u,v,w) the element of R  as
(1)Φn(u,v,w)=fn(uvw)-∑i+j+k=n‍fi(σn-i(u))dj(σkτi(v))dk(τn-k(w)).
Then, by using similar arguments as used in Lemma 2.2(iii) of [8], it can easily be seen that Φn(u,v,w) is additive in each argument and Φn(u,v,w)=-Φn(w,v,u).Following lemmas are essential for developing the proof of our main result. Proof of Lemma1 can be seen in [15] whereas Lemma 2 was obtained in [9].Lemma 1.
Assume thatR is a prime ring of characteristic different from two and U is a noncentral square closed Lie ideal of R. Let G1,G2,…,Gn be additive groups, S:G1×G2×⋯×Gn→R and let T:G1×G2×⋯×Gn→R be the mappings which are additive in each argument. If S(a1,a2,…,an)xT(a1,a2,…,an)={0} for every x∈U, ai∈Gi, i=1,2,…,n, then S(a1,a2,…,an)={0} for every ai∈Gi, i=1,2,…,n or T(b1,b2,…,bn)={0} for every bi∈Gi, i=1,2,…,n.Lemma 2.
LetR be a prime ring of characteristic different from two and let U be a square closed Lie ideal of R such that U⊈Z(R). Then there exist elements u,v,w∈U such that [u,v,w]≠0.Now, we prove the following lemma which plays a crucial role in developing the proof of our main result.Lemma 3.
LetR be a 2-torsion free ring and let U be a square closed Lie ideal of R. If F={fn}n∈ℕ is a generalized Jordan triple (σ,τ)-higher derivation of U into R and Φm(u,v,w)=0 for all u,v,w∈U and every m<n, then Φn(u,v,w)τn(x)τn[w,v,u]=0 for all u,v,w,x∈U and every n∈ℕ.Proof.
Letζ=4(uvwxwvu+wvuxuvw). Then
(2)fn(ζ)=fn(u(4vwxwv)u)+fn(w(4vuxuv)w)=4∑i+j+k=n‍fi(σn-i(u))×dj(σkτi(v(wxw)v))dk(τn-k(u))+4∑i+j+k=n‍fi(σn-i(w))×dj(σkτi(v(uxu)v))dk(τn-k(w))=4∑i+l+y+p+q+s+k=n‍fi(σn-i(u))×dl(σy+p+q+s+kτi(v))dy(σp+q+s+kτi+l(w))×dp(σq+s+kτi+l+y(x))dq(σs+kτi+l+y+p(w))×ds(σkτi+l+y+p+q(v))dk(τn-k(u))+4∑i+l+y+p+q+s+k=n‍fi(σn-i(w))×dl(σy+p+q+s+kτi(v))×dy(σp+q+s+kτi+l(u))×dpσq+s+kτi+l+y(x)dq(σs+kτi+l+y+p(u))×ds(σkτi+l+y+p+q(v))dk(τn-k(w)).
From our hypothesis we have(3)fn(uvu)=∑i+j+k=n‍fi(σn-i(u))dj(σkτi(v))dk(τn-k(u)),BBBBBBBBBBBBBBBBBBBBBBBBBB∀u,v∈U,n∈ℕ.
Linearizing the above equation we obtain(4)fn(uvw+wvu)=∑i+j+k=n‍fi(σn-i(u))dj(σkτi(v))dk(τn-k(w))+∑i+j+k=n‍fi(σn-i(w))dj(σkτi(v))dk(τn-k(u))mmmmmmmmmmmmmmmmlm∀u,v∈U,n∈ℕ.
Again considerfn(ζ)=fn((2uvw)x(2wvu)+(2wvu)x(2uvw)). Using relation (4) we have
(5)fn(ζ)=4∑α+p+γ=n‍fα(σn-α(uvw))×dp(σγτα(x))dγ(τn-γ(wvu))+fα(σn-α(wvu))dp(σγτα(x))dγ(τn-γ(uvw)).
Combining the two equations (2) and (5)
(6)0=4∑i+l+y+p+q+s+k=n‍fi(σn-i(u))dl(σy+p+q+s+kτi(v))×dy(σp+q+s+kτi+l(w))×dp(σq+s+kτi+l+y(x))dq(σs+kτi+l+y+p(w))×ds(σkτi+l+y+p+q(v))dk(τn-k(u))-4∑α+p+γ=n‍fα(σn-α(uvw))dp(σγτα(x))×dγ(τn-γ(wvu))+4∑i+l+y+p+q+s+k=n‍fi(σn-i(w))dl(σy+p+q+s+kτi(v))×dy(σp+q+s+kτi+l(u))×dp(σq+s+kτi+l+y(x))dq(σs+kτi+l+y+p(u))×ds(σkτi+l+y+p+q(v))dk(τn-k(w))-4∑α+p+γ=n‍fα(σn-α(wvu))dp(σγτα(x))×dγ(τn-γ(uvw)).
Considering the first term,(7)∑i+l+y+p+q+s+k=n‍fi(σn-i(u))dl(σy+p+q+s+kτi(v))×dy(σp+q+s+kτi+l(w))dp(σq+s+kτi+l+y(x))×dq(σs+kτi+l+y+p(w))ds(σkτi+l+y+p+q(v))dk(τn-k(u))=∑i+l+y+q+s+k=n‍fi(σn-i(u))dl(σy+q+s+kτi(v))×dy(σq+s+kτi+l(w))(σq+s+kτi+l+y(x))×dq(σs+kτi+l+y(w))ds(σkτi+l+y+q(v))dk(τn-k(u))+∑i+l+y+q+s+k=n-1‍fi(σn-i(u))dl(σy+1+q+s+kτi(v))×dy(σ1+q+s+kτi+l(w))×d1(σq+s+kτi+l+y(x))dq(σs+kτi+l+y+1(w))×ds(σkτi+l+y+1+q(v))dk(τn-k(u))+⋯+∑i+l+y+q+s+k=1‍fi(σn-i(u))dl(σy+n-1+q+s+kτi(v))×dy(σn-1+q+s+kτi+l(w))×dn-1(σq+s+kτi+l+y(x))dq(σs+kτi+l+y+n-1(w))×ds(σkτi+l+y+n-1+q(v))dk(τn-k(u))+∑i+l+y+q+s+k=0‍fi(σn-i(u))dl(σy+n+q+s+kτi(v))×dy(σn+q+s+kτi+l(w))×dn(σq+s+kτi+l+y(x))×dq(σs+kτi+l+y+n(w))ds(σkτi+l+y+n+q(v))dk(τn-k(u))=∑i+l+y=n‍fi(σn-i(u))dl(σyτi(v))×dy(τn-y(w))τn(x)τn(wvu)+∑q+s+k=n‍σn(uvw)σn(x)dq(σn-q(w))×ds(σkτq(v))dk(τn-k(u))+∑0<i+l+y,q+s+k≤n-1i+l+y+q+s+k=n‍fi(σn-i(u))dl(σy+q+s+kτi(v))×dy(σq+s+kτi+l(w))(σq+s+kτi+l+y(x))×dq(σs+kτi+l+y(w))ds(σkτi+l+y+q(v))dk(τn-k(u))+∑i+l+y+q+s+k=n-1‍fi(σn-i(u))dl(σy+q+s+k+1τi(v))×dy(τi+lσq+s+k+1(w))×d1(σq+s+kτi+l+y(x))dq(σs+kτi+l+y+1(w))×ds(σkτi+l+y+q+1(v))dk(τn-k(u))+⋯+f1(σn-1(u))(σn-1τ(v))(σn-1τ(w))×dn-1(τ(x))τn(wvu)+σn(u)d1(σn-1(v))(σn-1τ(w))×dn-1(τ(x))τn(wvu)+σn(uv)d1(σn-1(w))dn-1(τ(x))τn(wvu)+σn(uvw)dn-1(σ(x))d1(τn-1(w))τn(vu)+σn(uvw)dn-1(σ(x))(στn-1(w))d1(τn-1(v))τn(u)+σn(uvw)dn-1(σ(x))(στn-1(w))×(στn-1(v))d1(τn-1(u))+σn(uvw)dn(x)τn(wvu).
Calculating the second term,(8)∑α+p+γ=n‍fα(σn-α(uvw))dp(σγτα(x))×dγ(τn-γ(wvu))=∑α+γ=n‍fα(σn-α(uvw))(σγτα(x))×dγ(τn-γ(wvu))+∑α+γ=n-1‍fα(σn-α(uvw))d1(σγτα(x))×dγ(τn-γ(wvu))+⋯+∑α+γ=1‍fα(σn-α(uvw))dn-1(σγτα(x))×dγ(τn-γ(wvu))+∑α+γ=0‍fα(σn-α(uvw))dn(σγτα(x))×dγ(τn-γ(wvu))=fn(uvw)τn(x)τn(wvu)+σn(uvw)σn(x)dn(wvu)+∑0<α,γ≤n-1α+γ=n‍fα(σn-α(uvw))σγτα(x)×dγ(τn-γ(wvu))+∑α+γ=n-1‍fα(σn-α(uvw))d1(σγτα(x))×dγ(τn-γ(wvu))+⋯+f1(σn-1(uvw))dn-1(τ(x))τn(wvu)+σn(uvw)dn-1(σ(x))d1(τn-1(wvu))+σn(uvw)dn(x)τn(wvu).
Using the hypothesis thatΦm(u,v,w)=0 for all m<n, we have
(9)∑α+p+γ=n‍fα(σn-α(uvw))dp(σγτα(x))dγ(τn-γ(wvu))=fn(uvw)τn(x)τn(wvu)+σn(uvw)σn(x)dn(wvu)+∑0<i+l+y,q+s+k≤n-1i+l+y+q+s+k=n‍fi(σn-i(u))dl(σy+q+s+kτi(v))×dy(σq+s+kτi+l(w))σq+s+kτi+l+y(x)×dq(σs+kτi+l+y(w))ds(σkτi+l+y+q(v))dk(τn-k(u))+∑i+l+y+q+s+k=n-1‍fi(σn-i(u))dl(σy+q+s+k+1τi(v))×dy(σq+s+k+1τi+l(w))×d1(σq+s+kτi+l+y(x))dq(σs+kτi+l+y+1(w))×ds(σkτi+l+y+q+1(v))dk(τn-k(u))+⋯+(∑i+l+y=1‍fi(σ1-iσn-1(u))dl(σyτiσn-1(v))×dy(τ1-yσn-1(w))dn-1(τ(x))τn(wvu)∑i+l+y=1‍)+σn(uvw)dn-1(σ(x))×(∑q+u+k=1‍dq(σ1-qτn-1(w))×du(σkτqτn-1(v))dk(σ1-kτn-1(u))∑q+u+k=1‍)+σn(uvw)dn(x)τn(wvu)=fn(uvw)τn(x)τn(wvu)+σn(uvw)σn(x)dn(wvu)+∑0<i+l+y,q+s+k≤n-1i+l+y+q+s+k=n‍fi(σn-i(u))dl(σy+q+s+kτi(v))×dy(σq+s+kτi+l(w))σq+s+kτi+l+y(x)×dq(σs+kτi+l+y(w))ds(σkτi+l+y+q(v))dk(τn-k(u))+∑i+l+y+q+s+k=n-1‍fi(σn-i(u))dl(σy+q+s+k+1τi(v))×dy(τi+lσq+s+k+1(w))×d1(σq+s+kτi+l+y(x))dq(σs+kτi+l+y+1(w))×ds(σkτi+l+y+q+1(v))dk(τn-k(u))+⋯+σn(uvw)dn-1(σ(x))d1(τn-1(w))τn(vu)+σn(uvw)dn-1(σ(x))(στn-1(w))d1(τn-1(v))τn(u)+σn(uvw)dn-1(σ(x))(στn-1(w))(στn-1(v))×d1(τn-1(u))+f1(σn-1(u))(σn-1τ(v))(τσn-1(w))×dn-1(τ(x))τn(wvu)+σn(u)d1(σn-1(v))(σn-1τ(w))dn-1(τ(x))τn(wvu)+σn(uv)d1(σn-1(w))dn-1(τ(x))τn(wvu)+σn(uvw)dn(x)τn(wvu).
Now, subtracting the two terms so obtained we find that(10)fn(uvw)τn(x)τn(wvu)-∑i+l+s=n‍fi(σn-i(u))dl(σsτi(v))×ds(τn-s(w))τn(x)τn(wvu)={∑i+l+s=n‍fn(uvw)-∑i+l+s=n‍fi(σn-i(u))dl(σsτi(v))ds(τn-s(w))}×τn(x)τn(wvu)=Φn(u,v,w)τn(x)τn(wvu).
Similarly, the difference of the last two terms yields(11)∑i+l+y+p+q+s+k=n‍fi(σn-i(w))dl(σy+p+q+s+kτi(v))×dy(σp+q+s+kτi+l(u))dp(σq+s+kτi+l+s(x))×dq(σs+kτi+l+y+p(u))×ds(σkτi+l+y+p+q(v))dk(τn-k(w))-∑α+p+γ=n‍fα(σn-α(wvu))dp(σγτα(x))dγ(τn-γ(uvw))=Φn(w,v,u)τn(x)τn(uvw).
Thus, (6) reduces to
(12)4(+Φn(w,v,u)τn(x)τn(uvw)Φn(u,v,w)τn(x)τn(wvu)+Φn(w,v,u)τn(x)τn(uvw))=0.
Since the characteristic ofR is not two, we find that Φn(u,v,w)τn(x)τn[w,v,u]=0.Lemma 4.
LetR be a prime ring of characteristic different from 2 and let U be a noncentral square closed Lie ideal of R. If F={fn}n∈ℕ is a generalized Jordan triple (σ,τ)-higher derivation on U into R, then Φn(u,v,w)=0, for all u,v,w∈U and each n∈ℕ.Proof.
Trivially forn=0, Φ0(u,v,w)=0. By induction let us assume that Φm(u,v,w)=0, for every u,v,w∈U and for all m<n. By Lemma 3, we have Φn(u,v,w)τn(x)τn[w,v,u]=0 or τ-n(Φn(u,v,w))x[w,v,u]=0. Again, by Lemma 1, we have Φn(u,v,w)=0 for all u,v,w∈U or [w,v,u]=0 for all u,v,w∈U. If [w,v,u]=0 for all u,v,w∈U holds, then this contradicts Lemma 2. Hence, in either case Φn(u,v,w)=0 for all u,v,w∈U,n∈ℕ.Theorem 5.
LetR be a prime ring of characteristic different from 2 and let U be a square closed Lie ideal of R such that U⊈Z(R). Then, every generalized Jordan triple (σ,τ)-higher derivation of U into R is a generalized (σ,τ)-higher derivation of U into R.Proof.
LetF={fn}n∈ℕ be generalized Jordan triple (σ,τ)-higher derivation.
It can be easily seen thatf0=IR. By induction, assume that fm(uv)=∑i+j=mfi(σn-i(u))dj(τn-j(v)) holds for all m<n.
Foru,v,x∈U, take ξ=4uvxuv∈U and use Lemma 4 to get
(13)fn(ξ)=4fn(u(vxu)v)=4∑i+j+k=n‍fi(σn-i(u))dj(σkτi(vxu))dk(τn-k(v))=4∑i+s+p+q+k=n‍fi(σn-i(u))ds(σp+q+kτi(v))×dp(σq+kτi+s(x))dq(σkτs+p+i(u))×dk(τn-k(v))=4(∑i+s+q+k=n‍fi(σn-i(u))ds(σq+kτi(v))×d0(σq+kτi+s(x))dq(σkτs+i(u))dk(τn-k(v))+∑i+s+q+k=n-1‍fi(σn-i(u))ds(σ1+q+kτi(v))×d1(σq+kτi+s(x))×dq(σkτ1+s+i(u))dk(τn-k(v))+⋯+∑i+s+q+k=1‍fi(σn-i(u))×ds(σn-1+q+kτi(v))dn-1(σq+kτi+s(x))×dq(σkτs+n-1+i(u))dk(τn-k(v))+∑i+s+q+k=0‍fi(σn-i(u))ds(σn+q+kτi(v))×dn(σq+kτi+s(x))×dq(σkτn+s+i(u))dk(τn-k(v))∑i+s+q+k=n‍)=4(∑i+s=n‍fi(σn-i(u))ds(τn-s(v))(τn(xuv))+∑q+k=n‍(σn(uvx))dq(σn-q(u))dk(τn-k(v))+∑0<i+s,q+k≤n-1i+s+q+k=n‍fi(σn-i(u))×ds(σq+kτi(v))(σq+kτi+s(x))×dq(σkτs+i(u))dk(τn-k(v))+∑i+s+q+k=n-1‍fi(σn-i(u))ds(σ1+q+kτi(v))×d1(σq+kτi+s(x))dq(σkτ1+s+i(u))dk(τn-k(v))+⋯+f1(σn-1(u))(σn-1τ(v))×dn-1(τ(x))(τn(uv))+σn(u)d1(σn-1(v))dn-1(τ(x))(τn(uv))+σn(uv)dn-1(σ(x))d1(τn-1(u))(τn(v))+σn(uv)dn-1(σ(x))(στn-1(u))d1(τn-1(v))+σn(uv)dn(x)τn(uv)∑i+s=n‍).
On the other hand,(14)fn(ξ)=4fn((uv)x(uv))=4∑i+j+k=n‍fi(σn-i(uv))dj(σkτi(x))dk(τn-k(uv))=4(∑i+k=n‍fi(σn-i(uv))d0(σkτi(x))dk(τn-k(uv))+∑i+k=n-1‍fi(σn-i(uv))d1(σkτi(x))dk(τn-k(uv))+⋯+∑i+k=1‍fi(σn-i(uv))dn-1(σkτi(x))dk(τn-k(uv))+∑i+k=0‍fi(σn-i(uv))dn(σkτi(x))dk(τn-k(uv))=4(∑i+k=n‍σn(uv)σn(x)dn(uv)+fn(uv)τn(x)τn(uv)+∑0<i,k≤n-1i+k=n‍fi(σn-i(uv))(σkτi(x))dk(τn-k(uv))+∑i+k=n-1‍fi(σn-i(uv))d1(σkτi(x))dk(τn-k(uv))+⋯+f1(σn-i(uv))dn-1(τ(x))(τn(uv))+(σn(uv))dn-1(σ(x))d1(τn-1(uv))+(σn(uv))dn(x)τn(uv)∑i+k=n‍))=4(+(σn(uv))dn(x)dk(τn(uv))σn(uv)σn(x)dn(uv)+fn(uv)τn(x)τn(uv)+∑0<l+s,p+q≤n-1l+s+p+q=n‍fl(σn-l(u))ds(τlσq+p(v))×(σq+kτl+s(x))×dq(σpτl+s(u))dp(τn-p(v))+∑l+s+p+q=n-1‍fl(σn-l(u))ds(τlσq+p+1(v))×d1(σq+kτl+s(x))dq(σpτl+s+1(u))×dp(τn-p(v))+⋯+f1(σn-1(u))(σn-1τ(v))dn-1(τ(x))(τn(uv))+σn(u)d1(σn-1(v))dn-1(τ(x))(τn(uv))+(σn(uv))dn-1(σ(x))d1(τn-1(u))τn(v)+(σn(uv))dn-1(σ(x))(στn-1(u))d1(τn-1(v))+(σn(uv))dn(x)dk(τn(uv))).
Comparing both equations, reordering the indices and as characteristic ofR is different from two we finally obtain
(15){fn(uv)-∑i+j=n‍fi(σn-i(u))kkkk×dj(τn-j(v))∑i+j=n‍}τn(x)τn(uv)=0kkkkkkkkkkkkikkkkkkkkkkkk∀u,v∈U.
Asτ is one-one and onto implementing Lemma 1, it can be easily seen that fn(uv)-∑i+j=nfi(σn-i(u))dj(τn-j(v))=0 for all u,v∈U or u′v′=0 for all u′,v′∈U. Now if u′v′=0 for all u′,v′∈U, then U is commutative and hence using similar arguments as used in the second paragraph of the proof of Lemma 1.3 of [1] U is central, a contradiction. Therefore, fn(uv)-∑i+j=nfi(σn-i(u))dj(τn-j(v))=0 for all u,v∈U. This completes the proof of our theorem.Remark 6.
Forσ=τ=IR and fn=dn for each n∈ℕ our result reduces to the main result obtained by Ferrero and Haetinger [9].Corollary 7 ([9, Theorem 1.2]).
LetR be a prime ring of characteristic different from two and let U be a noncentral square closed Lie ideal of R. Then, every Jordan triple higher derivation of U into R is a higher derivation of U into R.



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*Source: 684792-2014-01-22.xml*