ISRN.ALGEBRA ISRN Algebra 2090-6293 Hindawi Publishing Corporation 365424 10.1155/2014/365424 365424 Research Article A Note on Jordan Triple Higher *-Derivations on Semiprime Rings Ezzat O. H. Aljadeff E. Jaballah A. Kılıçman A. Kittaneh F. You H. Mathematics Department Al-Azhar University Nasr City Cairo 11884 Egypt azhar.edu.eg 2014 942014 2014 09 02 2014 26 03 2014 9 4 2014 2014 Copyright © 2014 O. H. Ezzat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce the following notion. Let 0 be the set of all nonnegative integers and let D=(di)i0 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,yR and each 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,yR. 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 n2, R is said to be n-torsion free if, for xR,nx=0 implies x=0. An additive mapping xx* satisfying (xy)*=y*x* and (x*)*=x for all x,yR is called an involution and R is called a *-ring.

An additive mapping d:RR is called a derivation if d(xy)=d(x)y+yd(x) holds for all x,yR, and it is called a Jordan derivation if d(x2)=d(x)x+xd(x) for all xR. Every derivation is obviously a Jordan derivation and the converse is in general not true [1, Example 3.2.1]. An influential Herstein theorem  shows that any Jordan derivation on a 2-torsion free prime ring is a derivation. Later on, Brešar  has extended Herstein’s theorem to 2-torsion free semiprime rings. A Jordan triple derivation is an additive mapping d:RR satisfying d(xyx)=d(x)yx+xd(y)x+xyd(x) for all x,yR. 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  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:RR is called a *-derivation if d(xy)=d(x)y*+xd(y) holds for all x,yR, and it is called a Jordan *-derivation if d(x2)=d(x)x*+xd(x) holds for all xR. 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  that noncommutative prime *-rings do not admit nontrivial *-derivations. A Jordan triple *-derivation is an additive mapping d:RR with the property d(xyx)=d(x)y*x*+xd(y)x*+xyd(x) for all x,yR. It could easily be seen that any Jordan *-derivation on a 2-torsion free *-ring is a Jordan triple *-derivation [6, Lemma  2]. Vukman  has proved that any Jordan triple *-derivation on a 6-torsion free semiprime *-ring is a Jordan *-derivation.

Let 0 be the set of all nonnegative integers and let D=(di)i0 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 n0,dn(xy)=i+j=ndi(x)dj(y) (resp., dn(x2)=i+j=ndi(x)dj(x)) holds for all x,yR. The concept of higher derivations was introduced by Hasse and Schmidt . This interesting notion of higher derivations has been studied in both commutative and noncommutative rings; see, for example, . Clearly, every higher derivation is a Jordan higher derivation. Ferrero and Haetinger  have extended Herstein's theorem  for higher derivations on 2-torsion free semiprime rings. For an account of higher and Jordan higher derivations the reader is referred to . A family D=(di)i0 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,yR. Ferrero and Haetinger  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 . 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 0 be the set of all nonnegative integers and let D=(di)i0 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 n0, (1)dn(xy)=i+j=ndi(x)dj(y*i)x,yR;

a Jordan higher *-derivation of R if, for each n0, (2)dn(x2)=i+j=ndi(x)dj(x*i)xR;

a Jordan triple higher *-derivation of R if, for each n0, (3)dn(xyx)=i+j+k=ndi(x)dj(y*i)dk(x*i+j)x,yR.

Throughout this section, we will use the following notation.

Notation. Let D=(di)i0 be a Jordan triple higher *-derivation of a *-ring R. For every fixed n0 and each x,yR, 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,yR. 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,yR are such that xry+yrx=0 for all rR, then xry=yrx=0 for all rR. If R is semiprime, then xry=0 for all rR 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,yR are such that xr*y*+yrx=0 for all rR, then xy=yx=0.

Lemma 4.

Let D=(di)i0 be a Jordan triple higher *-derivation of a *-ring R. If Am(x)=0 for all xR and for each mn, then An(x)y*nx2*n+x2yAn(x)=0 for each n0 and for every x,yR.

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+x2yr+k=ndr(x)dk(x*r)+i+p+q+r+k=ni+pn,r+kndi(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=nin,kndi(x2)dj((y)*i)dk(x2*i+j)  =dn(x2)y*nx*nx*n+x2ydn(x2)+i+j+k=nin,kn(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+vn,s+tndu(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)i0 of R is a Jordan higher *-derivation of R.

Proof.

We intend to show that An(x)=0 for all xR. In case n=0, we get trivially A0(x)=0 for all xR. If n=1, then it follows from [7, Theorem 1] that A1(x)=0 for all xR. Thus we assume that Am(x)=0 for all xR and m<n. Thus, from Lemma 4, we see that (8)An(x)y*nx2*n+x2yAn(x)=0x,yR. 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=0xR,(10)x2An(x)=0xR. 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)=0x,yR. 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)=0x,yR. 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)=0x,yR. 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)=0x,yR. Subtracting the relation (13) from (14) we obtain, since R is 3-torsion free, that (15)Bn(x,y)x2+An(x)(xy+yx)=0x,yR. Right multiplication of (15) by An(x)x and using (9) we obtain (16)An(x)xyAn(x)x+An(x)yxAn(x)x=0x,yR. Putting yx for y in (16) and left-multiplying by x we get (xAn(x)x)y(xAn(x)x)=0, for all x,yR. By the semiprimeness of R it follows that xAn(x)x=0 for all xR. So (16) reduces to An(x)xyAn(x)x=0, for all x,yR. Again, by the semiprimeness of R, we get (17)An(x)x=0xR. Using (17), (15) reduces to Bn(x,y)x2+An(x)yx=0 for all x,yR. 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,yR. Again, by the semiprimeness of R, we get (18)xAn(x)=0xR. Linearizing (17) we have (19)An(x)y+Bn(x,y)x+An(y)x+Bn(x,y)y=0x,yR. Putting -x for x in (19) we get (20)An(x)y+Bn(x,y)x-An(y)x-Bn(x,y)y=0x,yR. Adding (19) and (20) we get, since R is 2-torsion free, that (21)An(x)y+Bn(x,y)x=0x,yR. Multiplying (21) by An(x) from the right and using (18) we get An(x)yAn(x)=0 for all x,yR. By the semiprimeness of R, we get An(x)=0 for all xR. 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)i0 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=nr+s=jdi(x)dr(x*i)ds(y*i+r)+i+j=nr+s=jdi(x)dr(y*i)ds(x*i+r)+i+j=nk+l=idk(x)dl(y*k)dj(x*k+l)+i+j=nk+l=idk(y)dl(x*k)dj(x*k+l)=i+r+s=ndi(x)dr(x*i)ds(y*i+r)+2i+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.

Ashraf M. Ali S. Haetinger C. On derivations in rings and their applications The Aligarh Bulletin of Mathematics 2006 25 2 79 107 MR2537802 Herstein I. N. Jordan derivations of prime rings Proceedings of the American Mathematical Society 1957 8 1104 1110 MR0095864 10.1090/S0002-9939-1957-0095864-2 Brešar M. Jordan derivations on semiprime rings Proceedings of the American Mathematical Society 1988 104 4 1003 1006 10.2307/2047580 MR929422 ZBL0691.16039 Herstein I. N. Topics in Ring Theory 1969 Chicago, Ill, USA The University of Chicago Press MR0271135 Brešar M. Jordan mappings of semiprime rings Journal of Algebra 1989 127 1 218 228 10.1016/0021-8693(89)90285-8 MR1029414 ZBL0691.16040 Brešar M. Vukman J. On some additive mappings in rings with involution Aequationes Mathematicae 1989 38 2-3 178 185 10.1007/BF01840003 MR1018911 ZBL0691.16041 Vukman J. A note on Jordan *—derivations in semiprime rings with involution International Mathematical Forum 2006 1 13–16 617 622 MR2251192 ZBL1143.16037 Hasse H. Schmidt F. K. Noch eine begrüdung der theorie der höheren differential quotienten in einem algebraaischen funktionenkorper einer unbestimmeten Journal für die Reine und Angewandte Mathematik 1937 177 215 237 Macarro L. N. On the modules of m-integrable derivations in non-zero characteristic Advances in Mathematics 2012 229 5 2712 2740 10.1016/j.aim.2012.01.015 MR2889143 Hoffmann D. Kowalski P. Integrating Hasse-Schmidt derivations 2012, http://arxiv.org/abs/1212.5788 Xiao Z. Wei F. Jordan higher derivations on triangular algebras Linear Algebra and its Applications 2010 432 10 2615 2622 10.1016/j.laa.2009.12.006 MR2608180 ZBL1185.47034 Wei F. Xiao Z. Generalized Jordan derivations on semiprime rings and its applications in range inclusion problems Mediterranean Journal of Mathematics 2011 8 3 271 291 10.1007/s00009-010-0081-9 MR2824581 ZBL1246.16033 Ferrero M. Haetinger C. Higher derivations and a theorem by Herstein Quaestiones Mathematicae. Journal of the South African Mathematical Society 2002 25 2 249 257 10.2989/16073600209486012 MR1916335 ZBL1009.16036 Haetinger C. Ashraf M. Ali S. On higher derivations: a survey International Journal of Mathematics, Game Theory, and Algebra 2011 19 5-6 359 379 MR2814896 ZBL1234.16030