We introduce the following notion. Let ℕ0 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.

1. Introduction

Let R be an associative ring, for any x,y∈R. Recall that R is prime if xRy=0 implies x=0 or y=0 and is semiprime 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 an involution and R is called a *-ring.

An additive mapping d:R→R is called a derivation if d(xy)=d(x)y+yd(x) holds for all x,y∈R, and it is called a Jordan 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. A Jordan 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.

Let R 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 a Jordan *-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. A Jordan 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.

Let ℕ0 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 a higher derivation (resp., a Jordan 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 a Jordan 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 of higher *-derivations, Jordan higher *-derivations, and Jordan 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.

Let ℕ0 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 higher *-derivation of R if, for each n∈ℕ0,
(1)dn(xy)=∑i+j=ndi(x)dj(y*i)∀x,y∈R;

a Jordan higher *-derivation of R if, for each n∈ℕ0,
(2)dn(x2)=∑i+j=ndi(x)dj(x*i)∀x∈R;

a Jordan 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 that An(-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 [<xref ref-type="bibr" rid="B3">5</xref>, Lemma 1.1]).

Let R 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 [<xref ref-type="bibr" rid="B12">7</xref>, Lemma 1]).

Let R 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.

Let D=(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 of xyx 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.

Let R 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 that An(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 [<xref ref-type="bibr" rid="B12">7</xref>, Theorem 1]).

Let R be a 6-torsion free semiprime *-ring. Then every Jordan triple *-derivation of R is a Jordan *-derivation of R.

Theorem 7.

Let R 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 Theorems 5 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.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is truly indebted to Professor M. N. Daif for his constant encouragement and valuable discussions. The author also would like to express sincere gratitude to the referees for their careful reading and helpful comments. This paper is a part of the author’s Ph.D. dissertation under the supervision of Professor M. N. Daif.

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