# A Note on Jordan Triple Higher *-Derivations on Semiprime Rings

Authors: O. H. Ezzat
Journal: ISRN Algebra
 (2014)

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## Abstract

We introduce the following notion. Letℕ0 be the set of all nonnegative integers and let D=(di)i∈ℕ0 be a family of additive mappings of a *-ring R such that d0=idR; D is called a Jordan higher *-derivation (resp., a Jordan higher *-derivation) of R if dn(x2)=∑i+j=n‍di(x)dj(x*i) (resp., dn(xyx)=∑i+j+k=n‍di(x)dj(y*i)dk(x*i+j)) for all x,y∈R and each n∈ℕ0. It is shown that the notions of Jordan higher *-derivations and Jordan triple higher *-derivations on a 6-torsion free semiprime *-ring are coincident.


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## Body

## 1. Introduction

LetR be an associative ring, for any x,y∈R. Recall that R isprime if xRy=0 implies x=0 or y=0 and issemiprime if xRx=0 implies x=0. Given an integer n≥2, R is said to be n-torsion free if, for x∈R,nx=0 implies x=0. An additive mapping x→x* satisfying (xy)*=y*x* and (x*)*=x for all x,y∈R is called aninvolution and R is called a *-ring.An additive mappingd:R→R is called aderivation if d(xy)=d(x)y+yd(x) holds for all x,y∈R, and it is called aJordan derivation if d(x2)=d(x)x+xd(x) for all x∈R. Every derivation is obviously a Jordan derivation and the converse is in general not true [1, Example 3.2.1]. An influential Herstein theorem [2] shows that any Jordan derivation on a 2-torsion free prime ring is a derivation. Later on, Brešar [3] has extended Herstein’s theorem to 2-torsion free semiprime rings. AJordan triple derivation is an additive mapping d:R→R satisfying d(xyx)=d(x)yx+xd(y)x+xyd(x) for all x,y∈R. Any derivation is obviously a Jordan triple derivation. It is also easy to see that every Jordan derivation of a 2-torsion free ring is a Jordan triple derivation [4, Lemma 3.5]. Brešar [5] has proved that any Jordan triple derivation of a 2-torsion free semiprime ring is a derivation.LetR be a *-ring. An additive mapping d:R→R is called a *-derivation if d(xy)=d(x)y*+xd(y) holds for all x,y∈R, and it is called aJordan*-derivation if d(x2)=d(x)x*+xd(x) holds for all x∈R. We might guess that any Jordan *-derivation of a 2-torsion free prime *-ring is a *-derivation, but this is not the case. It has been proved in [6] that noncommutative prime *-rings do not admit nontrivial *-derivations. AJordan triple*-derivation is an additive mapping d:R→R with the property d(xyx)=d(x)y*x*+xd(y)x*+xyd(x) for all x,y∈R. It could easily be seen that any Jordan *-derivation on a 2-torsion free *-ring is a Jordan triple *-derivation [6, Lemma2]. Vukman [7] has proved that any Jordan triple *-derivation on a 6-torsion free semiprime *-ring is a Jordan *-derivation.Letℕ0 be the set of all nonnegative integers and let D=(di)i∈ℕ0 be a family of additive mappings of a ring R such that d0=idR. Then D is said to be ahigher derivation (resp., aJordan higher derivation) of R if, for each n∈ℕ0,dn(xy)=∑i+j=n‍di(x)dj(y) (resp., dn(x2)=∑i+j=n‍di(x)dj(x)) holds for all x,y∈R. The concept of higher derivations was introduced by Hasse and Schmidt [8]. This interesting notion of higher derivations has been studied in both commutative and noncommutative rings; see, for example, [9–13]. Clearly, every higher derivation is a Jordan higher derivation. Ferrero and Haetinger [13] have extended Herstein's theorem [2] for higher derivations on 2-torsion free semiprime rings. For an account of higher and Jordan higher derivations the reader is referred to [14]. A family D=(di)i∈ℕ0 of additive mappings of a ring R, where d0=idR, is called aJordan triple higher derivation if dn(xyx)=∑i+j+k=n‍di(x)dj(yi)dk(xi+j) holds for all x,y∈R. Ferrero and Haetinger [13] have proved that every Jordan higher derivation of a 2-torsion free ring is a Jordan triple higher derivation. They also have proved that every Jordan triple higher derivation of a 2-torsion free semiprime ring is a higher derivation.Motivated by the notions of*-derivations and higher derivations, we naturally introduce the notions ofhigher*-derivations,Jordan higher*-derivations, andJordan triple higher*-derivations. Our main objective in this paper is to show that every Jordan triple higher *-derivation of a 6-torsion free semiprime *-ring is a Jordan higher *-derivation. This result extends the main result of [7]. It is also shown that every Jordan higher *-derivation of a 2-torsion free *-ring is a Jordan triple higher *-derivation. So we can conclude that the notions of Jordan triple higher *-derivations and Jordan higher *-derivations are coincident on 6-torsion free semiprime *-rings.


## 2. Preliminaries and Main Results

We begin by the following definition.Definition 1.
Letℕ0 be the set of all nonnegative integers and let D=(di)i∈ℕ0 be a family of additive mappings of a *-ring R such that d0=idR. D is called(a)
ahigher*-derivation of R if, for each n∈ℕ0,
(1)dn(xy)=∑i+j=n‍di(x)dj(y*i)∀x,y∈R;(b)
aJordan higher*-derivation of R if, for each n∈ℕ0,
(2)dn(x2)=∑i+j=n‍di(x)dj(x*i)∀x∈R;(c)
aJordan triple higher*-derivation of R if, for each n∈ℕ0,
(3)dn(xyx)=∑i+j+k=n‍di(x)dj(y*i)dk(x*i+j)∀x,y∈R.Throughout this section, we will use the following notation.Notation. Let D=(di)i∈ℕ0 be a Jordan triple higher *-derivation of a *-ring R. For every fixed n∈ℕ0 and each x,y∈R, we denote by An(x) and Bn(x,y) the elements of R defined by
(4)An(x)=dn(x2)-∑i+j=n‍di(x)dj(x*i),Bn(x,y)=dn(xy+yx)-∑i+j=n‍di(x)dj(y*i)-∑i+j=n‍di(y)dj(x*i).It can easily be seen thatAn(-x)=An(x), Bn(-x,y)=-Bn(x,y), and An(x+y)=An(x)+An(y)+Bn(x,y) for each pair x,y∈R. We will use these relations without any explicit mention in the steps of the proofs. The next lemmas are crucial in developing the proofs of the main results.Lemma 2 (see [5, Lemma 1.1]).
LetR be a 2-torsion free semiprime ring. If x,y∈R are such that xry+yrx=0 for all r∈R, then xry=yrx=0 for all r∈R. If R is semiprime, then xry=0 for all r∈R implies yrx=xy=yx=0.Lemma 3 (see [7, Lemma 1]).
LetR be a 2-torsion free semiprime *-ring. If x,y∈R are such that xr*y*+yrx=0 for all r∈R, then xy=yx=0.Lemma 4.
LetD=(di)i∈ℕ0 be a Jordan triple higher *-derivation of a *-ring R. If Am(x)=0 for all x∈R and for each m≤n, then An(x)y*nx2*n+x2yAn(x)=0 for each n∈ℕ0 and for every x,y∈R.Proof.
The substitution ofxyx for y in the definition of Jordan triple higher *-derivation gives
(5)dn(x(xyx)x)=∑i+j+k=n‍di(x)dj((xyx)*i)dk(x*i+j)=∑i+j+k=n‍di(x)(∑p+q+r=j‍dp(x*i)dq(y*i+p)dr(x*i+p+q))ccccccccc×dk(x*i+j)=∑i+p+q+r+k=n‍di(x)dp(x*i)dq(y*i+p)dr(x*i+p+q)cccccccccccc×dk(x*i+p+q+r)=∑i+p=n‍di(x)dp(x*i)y*nx*nx*n+x2y∑r+k=n‍dr(x)dk(x*r)+∑i+p+q+r+k=ni+p≠n,r+k≠ndi(x)dp(x*i)dq(y*i+p)dr(x*i+p+q)ccccccccccccccc×dk(x*i+p+q+r).
On the other hand, the substitution of x2 for x in the definition of Jordan triple higher *-derivation and using our assumption that Am(x)=0 for m<n give
(6)dn(x2yx2)=∑i+j+k=n‍di(x2)dj((y)*i)dk(x2*i+j)=dn(x2)y*nx*nx*n+x2ydn(x2)+∑i+j+k=ni≠n,k≠ndi(x2)dj((y)*i)dk(x2*i+j)=dn(x2)y*nx*nx*n+x2ydn(x2)+∑i+j+k=ni≠n,k≠n(∑u+v=i‍du(x)dv(x*u))dj(y*i)cccccccccccc×(∑s+t=k‍ds(x*i+j)dt(x*i+j+s))=dn(x2)y*nx*nx*n+x2ydn(x2)+∑u+v+j+s+t=nu+v≠n,s+t≠ndu(x)dv(x*u)dj(y*u+v)dscccccccccccccccccc×(x*u+v+j)dt(x*u+v+j+s).
Now, subtracting the two relations so obtained we find that
(7)(dn(x2)-∑i+p=n‍di(x)dp(x*i))y*nx2*n+x2y(dn(x2)-∑r+k=n‍dr(x)dk(x*r))=0.
Using our notation the last relation reduces to the required result.Now, we are ready to prove our main results.Theorem 5.
LetR be a 6-torsion free semiprime *-ring. Then every Jordan triple higher *-derivation D=(di)i∈ℕ0 of R is a Jordan higher *-derivation of R.Proof.
We intend to show thatAn(x)=0 for all x∈R. In case n=0, we get trivially A0(x)=0 for all x∈R. If n=1, then it follows from [7, Theorem 1] that A1(x)=0 for all x∈R. Thus we assume that Am(x)=0 for all x∈R and m<n. Thus, from Lemma 4, we see that
(8)An(x)y*nx2*n+x2yAn(x)=0∀x,y∈R.
In case n is even, (8) reduces to An(x)yx2+x2yAn(x)=0; by applying Lemma 2 we get An(x)x2=x2An(x)=0. In case n is odd, (8) reduces to An(x)y*x2*+x2yAn(x)=0; by applying Lemma 3 we get An(x)x2=x2An(x)=0. So for either of the two cases we have for each n(9)An(x)x2=0∀x∈R,(10)x2An(x)=0∀x∈R.
The substitution of x+y for x in relation (9) gives
(11)An(x)y2+An(y)x2+Bn(x,y)x2+Bn(x,y)y2+An(x)(xy+yx)+An(y)(xy+yx)+Bn(x,y)(xy+yx)=0∀x,y∈R.
Substituting -x for x in (11) we obtain
(12)An(x)y2+An(y)x2-Bn(x,y)x2-Bn(x,y)y2-An(x)(xy+yx)-An(y)(xy+yx)+Bn(x,y)(xy+yx)=0∀x,y∈R.
Comparing (11) and (12) we get, since R is 2-torsion free, that
(13)Bn(x,y)x2+Bn(x,y)y2+An(x)(xy+yx)+An(y)(xy+yx)=0∀x,y∈R.
Putting 2x for x in (13) gives by the assumption that R is 2-torsion free that
(14)4Bn(x,y)x2+Bn(x,y)y2+4An(x)(xy+yx)+An(y)(xy+yx)=0∀x,y∈R.
Subtracting the relation (13) from (14) we obtain, since R is 3-torsion free, that
(15)Bn(x,y)x2+An(x)(xy+yx)=0∀x,y∈R.
Right multiplication of (15) by An(x)x and using (9) we obtain
(16)An(x)xyAn(x)x+An(x)yxAn(x)x=0∀x,y∈R.
Putting yx for y in (16) and left-multiplying by x we get (xAn(x)x)y(xAn(x)x)=0, for all x,y∈R. By the semiprimeness of R it follows that xAn(x)x=0 for all x∈R. So (16) reduces to An(x)xyAn(x)x=0, for all x,y∈R. Again, by the semiprimeness of R, we get
(17)An(x)x=0∀x∈R.
Using (17), (15) reduces to Bn(x,y)x2+An(x)yx=0 for all x,y∈R. Multiplying this relation by A(x) from the right and by x from the left we get xAn(x)yxAn(x)=0 for all x,y∈R. Again, by the semiprimeness of R, we get
(18)xAn(x)=0∀x∈R.
Linearizing (17) we have
(19)An(x)y+Bn(x,y)x+An(y)x+Bn(x,y)y=0∀x,y∈R.
Putting -x for x in (19) we get
(20)An(x)y+Bn(x,y)x-An(y)x-Bn(x,y)y=0∀x,y∈R.
Adding (19) and (20) we get, since R is 2-torsion free, that
(21)An(x)y+Bn(x,y)x=0∀x,y∈R.
Multiplying (21) by An(x) from the right and using (18) we get An(x)yAn(x)=0 for all x,y∈R. By the semiprimeness of R, we get An(x)=0 for all x∈R. This completes the proof of the theorem.Corollary 6 (see [7, Theorem 1]).
LetR be a 6-torsion free semiprime *-ring. Then every Jordan triple *-derivation of R is a Jordan *-derivation of R.Theorem 7.
LetR be a 2-torsion free *-ring. Then every Jordan higher *-derivation D=(di)i∈ℕ0 of R is a Jordan triple higher *-derivation of R.Proof.
We have(22)dn(x2)=∑i+j=n‍di(x)dj(x*i).
Put v=x+y and using (22) we obtain
(23)dn(v2)=∑i+j=n‍di(x+y)dj((x+y)*i)=∑i+j=n‍(di(x)dj(x*i)+di(y)dj(y*i)+di(x)dj(y*i)+di(y)dj(x*i)),dn(v2)=dn(x2+xy+yx+y2)=dn(x2)+dn(y2)+dn(xy+yx)=∑l+m=n‍dl(x)dm((x)*l)+∑r+s=n‍dr(y)ds((y)*r)+dn(xy+yx).
Comparing the last two forms of dn(v2) gives
(24)dn(xy+yx)=∑i+j=n‍(di(x)dj(y*i)+di(y)dj(x*i)).
Now put w=x(xy+yx)+(xy+yx)x. Using (24) we get
(25)dn(w)=∑i+j=n‍di(x)dj((xy+yx)*i)+∑i+j=n‍di(xy+yx)dj(x*i)=∑i+j=n‍∑r+s=j‍di(x)dr(x*i)ds(y*i+r)+∑i+j=n‍∑r+s=j‍di(x)dr(y*i)ds(x*i+r)+∑i+j=n‍∑k+l=i‍dk(x)dl(y*k)dj(x*k+l)+∑i+j=n‍∑k+l=i‍dk(y)dl(x*k)dj(x*k+l)=∑i+r+s=n‍di(x)dr(x*i)ds(y*i+r)+2∑i+j+k=n‍di(x)dj(y*i)dk(x*i+j)+∑k+l+j=n‍dk(y)dl(x*k)dj(x*k+l).
Also,
(26)dn(w)=dn((x2y+yx2)+2xyx)=dn(x2y+yx2)+2dn(xyx)=2dn(xyx)+∑r+s+j=n‍dr(x)ds(x*r)dj(y*r+s)+∑i+k+l=n‍di(y)dk(x*i)dl(x*i+k).
Comparing the last two forms of dn(w) and using the fact that R is 2-torsion free, we obtain the required result.By Theorems5 and 7, we can state the following.Theorem 8.
The notions of Jordan higher*-derivation and Jordan triple higher *-derivation on a 6-torsion free semiprime *-ring are coincident.



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