Jordan triple (α, β) higher * derivations on Semi prime rings
Dr.D.BHARATHI 1, P.RAVI 2
1
Associative Professor, Department of Mathematics, S.V.University Tirupati – A.P, India
E-Mail:bharathikavali@yahoo.co.in
2
Research Scholar, Department of Mathematics, S.V.University Tirupati– AP, India
E-Mail: rv1keerthana@gmail.com
ABSTRACT: In this paper we Prove, if R is a 6-torsion free semi prime * ring, then every Jordan triple
 ,   higher * derivation D  d i iN
0
of R is a Jordan  ,   higher * derivation of R. and also prove
that let R be a 2-torsion free * ring, then every Jordan  ,   higher * derivation D  d i iN of R is a
0
Jordan triple  ,   higher * derivation of R.
Key words: Higher derivation, Jordan higher derivation, triple  ,   higher * derivation,  ,  
higher derivation,  ,   higher * derivation, Jordan triple  ,   higher * derivation .
1. Introduction
Let R be an associative ring .For any x, y  R , R is said to be prime if xRx  0 implies x=0 or y=0 and
is Semi prime 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 . Let N 0 be the set of all nonnegative integers and let D  d i iN0 be a
family of additive mappings of a ring R such that d 0  id R . Then D is said to be higher derivation (resp. a
Jordan higher derivation ) of R if ,for n  N 0 , d n  xy 
 d x d  y 
i  j n
i
j
   d x d x ) holds for x, y  R. the concept of higher derivations was introduced by
(resp. d n x 2 
i  j n
i
j
Hasse and Schmidt [8]].This interesting notion of higher derivations have been studied in both
commutative and non commutative rings. Clearly, every higher derivation is a Jordan higher derivation.
Ferrero and Haetinger [10] have extended Herstein’s theorem [2] for higher derivation on 2-torsion free
semi prime rings. For an account of higher and Jordan higher derivations the reader is referred to [11].A
family D  d i iN of additive mappings of a ring R, where d 0  id R is called a Jordan triple higher
0
derivation if d n xyx 
 d xd  y  d x
i j
i
i  j  k n
i
j
k
holds for all x, y  R . Ferrero and Haetinger [10] 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 semi prime ring is a
higher derivation.
Motivated by the notions of *-derivation and higher derivations, we naturally introduce the notions of
higher * - derivations, Jordan higher *-derivations,  ,   higher *-derivations , Jordan triple higher *-
 ,   higher * - derivations .Our main objective in this paper is to show that
every Jordan triple  ,   higher * - derivation of a 6-torsion free semi prime *- ring is a Jordan  ,  
derivations, Jordan triple
higher *-derivation. This result extends the main result of [12].It is also shown that every Jordan higher
 ,   *-derivation of a 2-torsion free *-ring is a Jordan triple  ,   higher *-derivation. So we
conclude that the notions of Jordan triple  ,   higher * - derivations and Jordan higher  ,   * derivations are coincident on 6-torsion free semi prime *-rings.
2. Preliminaries and Main Results:
We begin by the following definition.
Definition: Let N 0 be the set of all nonnegative integers and let D  d iN 0 be a family of additive
mappings of a *-ring R such that d 0  id R . D is called
(a)  ,   Higher derivation of R if, for each n  N 0 ,
d n xy 
 d  x d   y 
n i
i  j n
i
n j
j
x, y  R
(1)
(b)  ,   Higher * - derivation
d n  xy 
 d  x d   y  
n i
i  j n
i
n j
* i
x, y  R
j
(2)
(C) Jordan  ,   higher * derivations
   d  x d  x  
dn x2 
n i
i  j n
i
n j
* i
j
(D) Jordan triple  ,   higher *-derivation
d n  xyx 
 d  x d 
n i
i  j  k n
i
j
k
 
 i  y * d k  n  k x *
i

i j
x, y  R
(3)
Throughout this section, we use the following notations
Notation: Let D  d i iN be a Jordan triple  ,   higher * - derivation of a *-ring R .For every fixed
0
n  N 0 and each x, y  R, we denote by An x  and Bn x, y  the elements of R defined by
   d  x d  x  
An x   d n x 2 
n i
i  j n
* i
n j
i
j
Bn  x, y   d n  xy  yx  
 d  x d   y     d   y d  x  
n i
i  j n
* i
n j
i
n i
j
i
i  j n
n j
* i
j
(4)
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 proof of the main results.
Lemmas 1 (see [5, Lemma 1.1]).Let R be a 2-torsion free semi prime ring .If x, y  R are such that
xry  yrx  0 for all r  R ,then xry  yrx  0 .If R is semi prime, then xry  0 for all r  R implies
yrx  xy  yx  0
Lemma 2(see [7, Lemma 1]).Let R be a 2-torsion free semi prime *-ring .If x, y  R are such that
xr* y *  yrx  0 for all r  R, then xy  yx  0
Lemma 3: Let D  d i iN be a Jordan triple  ,   higher * derivation of a * ring R .if Am x   0 for
0
all x  R and for each m  n , then An x  y * x 2
n
*n
 x 2 yAn x   0 for each n  N 0 and for every
x, y  R .
Proof: Consider Jordan triple  ,   higher * derivation
d n  xyx 
 d 
i  j k n
n i
i
 
 
( x) d j  k  i ( y ) * d k  n  k ( x) *
i

i j
x, y  R
The substitution of xyx for y in the above equation it gives
d  x xyxx  
 d 
i  j  k n
=
i
n i
 
( x) d j  k  i  xyx
 d  x d
n i
i  j k n
i
nk
k
  x  y  x  d 
k
j
 d 
* i
i
* i
* i
* i
k
x * 
i j
nk
x 
i j
*
By hypothesis, we have

 


i p
* i
i
d i  ni x    d p  k  i n p x  d q  k  i r  p  y  d r  k  i  nr x 
i  j  k n
 pqr  j





i pq


  d k  nk x *



i j

 d  x d  x   y
n i
i  p n
* i
n p
i
*n
p
n
 d  x d  x  
n
nr
x* x*  x 2 y
r  k n

* r
nk
r

k
 


i p
* i
i
d i  ni x    d p  k  i n p x  d q  k  i r  p  y  d r  k  i  nr x 
i  j  k n
 pqr  j





i pq


  d k  nk x *


(5)
On the other hand, the substitution of x 2 for x in the definition of Jordan triple  ,   higher *derivation and using our assumption that Am x   0 for m  n
  d  x d 

d n x 2 yx 2 

n i
i  j  k n

2
i
k
j
 
n
 
 
*n
*n
 dn x y x x
nk
i  j  k n
i  j ,k  n
2
j
k
 

2

 i y *  d k  nk x 2 
i

 x yd n x     d u  n i n u  x  d v  n i  n u x *
i  j  k  n  u  v i
2
 
n
s
 

* i j
  d 
u
k
j
  y 
i j
n

 

i js
 
n


d s  n  s n  s  x 
* u v j
 
i
*
i
 n  s x *  d t  n  k  n t x 
d n x 2 y * x * x *  x 2 yd n x 2 
=
n i
i
i  n ,k  n
 d 
s t  k
*n
x, y  R
   d  x d 
d n x 2 yx 2  d n x 2 y * x * x *  x 2 yd n x 2 
2
* i j
i
n
n

 i  y * d k  nk x 2 
n
n

 d  x d 
n u
u
u v  j  s t n
u v  n,s t  n
d t  nk x 
 d n x 2 y * x * x *  x 2 yd n x 2 

 
u v j
d s  nk  ns x *
 
 u x * d j  k  i  y * 

i u v
u


* u v j  s
 
n
k
v

 d 
n i
u
u v  j  s t n
u  v  n , s t  n
 n u x d v  n i k  u x *  d j  k  i y * 
i u v
u
(6)

d t  n  k  n t  x 
u v s  j
Now subtracting (5) from (6) and reordering indices, we get

 d n x 2   d i  n i  x  d p  n  p x *

i  p n

 

 
   y
i

*n
x2
*n
 


* r 
 x 2 y d n x 2   d r  n  r  x  d k  n  k  x    0
r  k n


 

(7)
An x  y * x 2
n
*n
 x 2 yAn x   0
Theorem 4: Let R be a 6-torsion free Semi prime * ring, then every Jordan triple  ,   higher *
derivation D  d i iN of R is a Jordan  ,   higher * derivation of R.
0

i pqr
Proof. We assume that D  d i iN is a Jordan triple  ,   higher * derivation
0
We have to show that An x   0
x  R .
In case n=0, we get trivially A0 x   0
x  R .
If n=1, then it follows from [7, Theorem 1] that A1 x   0
x  R .Thus we assume that Am x   0
for all x  R and m<n.
Thus by lemma (3), we see that
An x  y * x 2
n
*n
 x 2 yAn x   0 ,
x, y  R
(8)
If n is even, equation (8) reduces to An  x  yx 2  x 2 yAn  x   0
By lemma (1) we get An  x x 2  x 2 An  x   0
If n is odd, equation (8) reduces to An x  y * x 2
n
*n
 x 2 yAn x   0
By applying lemma (2) , we get
An x x 2  x 2 An x   0
So for either of the two cases, we have for each n
An x x 2  0
x  R
(9)
x 2 An  x   0
x  R
(10)
The substitution of x+y for x in relation (9) we have
An x  y x  y   0
2
x, y  R
An x  y 2  An  y x 2  Bn x, y x 2  Bn x, y  y 2  An x xy  yx   An  y xy  yx   Bn x, y xy  yx   0
x, y  R
(11)
Substituting -x for x in equation (11) we obtain
An x  y 2  An  y x 2  Bn x, y x 2  Bn x, y  y 2  An x xy  yx   An  y xy  yx   Bn x, y xy  yx   0
x, y  R
(12)
Comparing equation (11) and (12) we get, since R is 2-torsion free, that
Bn x, y x 2  Bn x, y  y 2  An x xy  yx   An  y xy  yx   0
(13)
Substitute 2x for x in equation (13) gives by the assumption that R is 2-torsion free that
4 Bn x, y x 2  Bn x, y  y 2  4 An x xy  yx   2 An  y xy  yx   0
4 Bn x, y x 2  Bn x, y  y 2  4 An x xy  yx   An  y xy  yx   0
(14)
Subtracting the equation (13) from (14) we obtain, since R is 3-torsion free, that
Bn x, y x 2  An x xy  yx   0
(15)
Right multiplication of (15) by
An x x
Bn x, y x 2 An x x  An x xy  yx An x x  0
BY using equation (10) we get
An x xyAn x x  An x yxAn x x  0
x, y  R
(16)
Substitute yx for y in (16) and left multiplying by x we get
xAn x x yxAn x x   0,
x, y  R .
By the semiprimeness of R it follows that
xAn x x  0
x, y  R .
So (16) reduces to
An x xyAn x x  0 ,
x, y  R
Again by using semiprimeness of R , we have
An x x  0
x  R
(17)
Using (17), (15) reduces to
Bn x, y x 2  An x  yx  0
x, y  R
Multiplying this relation by An x  from the right and by x from the left we get
xAn x yx An x   0
x, y  R .
Again, by the semiprimeness of R we get
xAn x   0
x, y  R
(18)
Linearizing (17) we have
An x y  Bn x, y x  An  y x  Bn x, y y  0
x, y  R
(19)
Putting –x for x in (19) we get
An x y  Bn x, y x  An  y x  Bn x, y y  0
(20)
Adding (19) and (20) we get, since R is 2-torsion free, that
2( An x  y  Bn x, y x)  0
x, y  R
( An x  y  Bn x, y x)  0
(21)
Multiplying (21) by An x  from the right and using (18) we get
An x  yAn x   0
x, y  R
By the semiprimeness of R , we get An x   0
x  R
i.e D  d i iN0 is a Jordan  ,   higher * derivations
This completes the proof of the theorem.
Corollary 5: (see [7, theorem1]).Let R be a 6-torsion free semi prime * ring .Then every Jordan
(α ,β) triple higher * derivation of R is a Jordan (α, β) higher * derivation of R.
Theorem 6. Let R be a 2-torsion free * ring .Then every Jordan (α , β) higher * derivation D  d i iN0
of R is a Jordan triple (α, β) higher * derivation of R .
Proof: By the definition of Jordan (α, β) higher * derivation
   d  x d
n i
d x2 
i  j n
i
j

n j
Put v  x  y in equation (22)
x  
* i
x  R
(22)
   d  x  y d  x  y  
n i
d v2 
i  j n
j
 d  x d
n i
i  j n
* i
n j
i
i
j

n j
x   d   y d
* i

n i
i
j
n j
y   d  x d
* i
n i
i
j

n j
y   d   y d
* i
n i
i
j

n j
x  
* i
(23)
 
 
 
d n v 2  d n x 2  d n y 2  d n xy  yx 
 d  x d
n l
l  mn
l
m

x     d   y d
* i
nm
nr
r
r  s n
j

ns
y    d xy  yx
* r
n
 
Comparing the last two forms of d n v 2 gives
d n  xy  yx  
 d  x d
n i
i
i  j n
j

n j
y    d   y d
* i
n i
i  j n
i
j

n j
x  
* i
(24)
Now put w  xxy  yx  xy  yxx using equation (24) ,we get

 

   d 
d n w   d i  ni x  d j  n j xy  yx 
d n w 
n i
i  j n r  s  j
   x d 
n i
k
i  j  n k  l i
d n w 

nk
i  j  n k  l i
n i
n j
j
n j
i

d r  nr x 
 

* k
 d 
* i
d k  n  k  y  d l  n i  x 

x 
s
i
n i
i  j n r  s  j

* i  k l
ns
xy  yxd j  n j x * 
   d 
ir
*
l
n i
 

d r  n r x  d s  n  j  y 
n l
  d  x 
i  j n r  s  j
n j
n i
i
i  j n
 
x d  y  d 
  d  x
i
* i

 
n i
i  j  n k  l i

x n j d r  nr y * d s  n j xir  



  d  x  
d k  n  k  y  d l  n l x *
i  j n r  s  j
* k l
j

Also d n w  d n x 2 y  yx 2  2 xyx


 d n x 2 y  yx 2 )  2d n ( xyx
 2d n xyx 
 
r  s n
  
n j
* ik 
j
 y i  r   2   d  ni x  n j d r  nr y * i d s  ns x i  r  
 d x 
* k
k
d r  nr x 2 d s  n s  y 
   d 
* r
i  k n
i
n i
 y d k  nk x 2 * 
i
 2d n xyx 

r  s  j n
 d   y 
n i
i  j k n
i
nk
nr
 

d r  nr x  d j  n j x 
 

 d 
* r
d k  n  k  x  d l  n l  x 
s
ns
 y * 
r j


* ik
Comparing the last two forms of d n w and using the fact that ‘R’ is 2-torsion free, we obtain the
required result.
i.e. D  d i iN is a Jordan triple  ,   higher * derivation
0
d n  xyx 
 d  x d 
n i
i  j  k n
i
j
k
 
 i  y * d k  n  k x *
i

i j
References:
[1] M.Ashraf, S.Ali, and C.Haetinger, “on derivations in rings and their applications”, The
Aligarh Bulletion of Mathematics, vol.25, no 2, pp 79-107, 2006.
[2] I.N.Herstein,” Jordan derivations of prime rings”, Proceedings of the American Mathematical
Society, vol.8, pp.1104-1110, 1957.
[3] M.Bresar,”Jordan derivations on semiprime rings’’, Proceedings of the American
Mathematical Society, vol,104, no 4, pp.1003-1006, 1988.
[4] I.N. Herstein, Topics in Ring Theory, the University of Chicago Press, Chicago, III, USA,
1969.
[5] M.Bresar, “Jordan mappings of semi prime rings “, Journal of Algebra, vol 127,no 1 pp.
218-228, 1989.
[6] M. Bresar and J.Vukman,”on some additive mappings in rings with involution”, Aequationes
Mathematicae, vol.38, no 2-3 pp. 178-185, 1989.
[7] J.Vukman,” A note on Jordan *-derivations in semi prime rings with involution,”
International Mathematical Forum, vol.1 no 13-16, pp.617-622, 2006.
[8] H.Hasse and F.K.Schmidt,” Noch eine begrudung der theorieder hoheren differential
Quotient in einem algebraaischen funktionenkorper einer unbestimmeten,” Journal fur die
Reineund Angewandte Mathematic, vol 177,pp 215-237,1937.
[9] L.N.Macarro, “on the modules of m-integrable derivations in non-zero characterstic,
Advances in Mathematics, vol 229, no.5 pp 2712-2740, 2012.
[10] M.Ferrero and C.Haetinger, “Higher derivations and theorem by Herstein, “Quaestiones
Mathematicae. Journal of the South African Mathematical Society, vol 25, no 2,pp 249257, 2002.
[11] C. Hettinger M.Ashraf, and S.Ali, “on higher derivations .a survey”, Internal Journal of
Mathematics, Game Theorey, and algebra, vol 19, no 5-6,pp 359-379, 2011.
[12] O.H.Ezzat “A Note on Jordan Triple Higher *- Derivations on Semiprime Rings”.Hindawi
Publishing Corporation, volume 2014, ISRN Algebra, 5 pages.
Profile of Authors:
Dr.D.Bharathi is working as associate Professor of Mathematics in the
Department of Mathematics, Sri Venkateswara University, Tirupati,
Andhrapradesh, India. She obtained her Ph.d from Sri Krishna Devaraya
University, Anantapur, AP, India. Her Research is focused on the topics
Associative and Non associative Rings, Latticetheory, Derivations on Rings,
Graph Theory, and Functional Analysis. She attended and organized several
conferences in Mathematics and her papers published in reputed journals. Her
teaching Experience is more than 15 years where she taught under graduate
and post graduate courses like B.sc., M.Sc. mathematics and applied
mathematics subjects like: Algebra, Real analysis, Partial differential equations,
Commutative Algebra, Mathematical Statistics, Functional analysis, Graph
theory, Galois theory etc, Integral equations to research scholars.
E-Mail:bharathikavali@yahoo.co.in
Mobile: 9440343888
P.RAVI, M.Sc., M. Phil., B.Ed, (Ph.D),
Research Scalar,
Deportment of Mathematics,
S.V.U, Tirupati
Email id: rv1keerthana@gmail.com
Mobile: 9704582404