Reprints From BAG2013

8/9/2019 Reprints From BAG2013

1/18

1 3

Beitrge zur Algebra und Geometrie /

Contributions to Algebra and

Geometry

Contributions to Algebra and Geometry

ISSN 0138-4821

Volume 54

Number 2

Beitr Algebra Geom (2013) 54:609-624

DOI 10.1007/s13366-012-0117-3

ordan left *-centralizers of prime andsemiprime rings with involution

Shakir Ali, Nadeem Ahmad Dar & Joso

Vukman


2/18

1 3

Your article is protected by copyright and all

rights are held exclusively by The Managing

Editors. This e-offprint is for personal use only

and shall not be self-archived in electronic

repositories. If you wish to self-archive yourarticle, please use the accepted manuscript

version for posting on your own website. You

may further deposit the accepted manuscript

version in any repository, provided it is only

made publicly available 12 months after

official publication or later and provided

acknowledgement is given to the original

source of publication and a link is inserted

to the published article on Springer's

website. The link must be accompanied by

the following text: "The final publication is

available at link.springer.com.


3/18

Beitr Algebra Geom (2013) 54:609624DOI 10.1007/s13366-012-0117-3

ORIGINAL PAPER

Jordan left -centralizers of prime and semiprime rings

with involution

Shakir Ali Nadeem Ahmad Dar Joso Vukman

Received: 27 February 2012 / Accepted: 22 June 2012 / Published online: 13 July 2012 The Managing Editors 2012

Abstract Let R be a ring with involution . An additive mapping T : R R is

called a left -centralizer (resp. Jordan left -centralizer) ifT(x y) = T(x)y (resp.

T(x2) = T(x)x) holds for allx,y R, and a reverse left-centralizer ifT(x y) =

T(y)x holds for all x,y R. In the present paper, it is shown that every Jordan left

-centralizer on a semiprime ring with involution, of characteristic different from two

is a reverse left -centralizer. This result makes it possible to solve some functional

equations in prime and semiprime rings with involution. Moreover, some more relatedresults have also been discussed.

Keywords Prime ring Semiprime ring Involution Left-centralizer

Reverse left-centralizer Reverse-centralizer Jordan left-centralizer

Mathematics Subject Classification (2000) 16N60 16W10

This research is partially supported by the Research Grants (UGC No. 39-37/2010(S R))

and (INT/SLOVENIA/P-18/2009).

S. Ali (B) N. A. Dar

Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India

e-mail: [email protected]; [email protected]

N. A. Dare-mail: [email protected]

J. Vukman

Department of Mathematics and Computer Science,

University of Maribor FNM, Koroska 160, 2000 Maribor, Slovenia

e-mail: [email protected]

1 3


4/18

610 Beitr Algebra Geom (2013) 54:609624

1 Introduction

This paper deals with the study of Jordan left -centralizers of prime and semi-

prime rings with involution , and was motivated by the work of Vukman(1997)

andZalar(1991). Throughout, Rwill represent an associative ring with centre Z(R).We shall denote byCthe extended centroid of a prime ring R. For the explanation of

Cwe refer the reader toMartindale(1969). Given an integern 2, a ring R is said

to be n-torsion free, if for x R,nx = 0 implies x = 0. As usual[x,y] and x y

will denote the commutator x y y xand anti-commutatorx y+ y x, respectively. An

additive mapping x x on a ring R is said to be an involution if(x y) = yx

and(x) = xholds forx,y R. A ring equipped with an involution is called a ring

with involution or -ring. Recall that R is prime if for a , b R,a Rb = (0)implies

a = 0 or b = 0, and is semiprime in case a Ra = (0) implies a = 0. Following

Zalar(1991), an additive mapping T : R R is called a left centralizer in caseT(x y) = T(x)y holds for all x,y R. For a semiprime ring R, all left centralizers

are of the form T(x) = q xfor all x R, where q is an element of Martindale right

ring of quotients QrofR(seeBeidar et al. 1996, Chapter 2 for details). In case Rhas

an identity element, thenT : R Ris a left centralizer if and only ifTis of the form

T(x) = ax for all x R and some fixed element a R. The definition of a right

centralizer should be self-explanatory. An additive mapping Tis called a two-sided

centralizer in caseT : R Ris a left and a right centralizer. In case T : R Ris a

two-sided centralizer, whereR is a semiprime ring with extended centoid C, then there

exists an element Csuch that T(x) = xfor allx R (viz;Beidar et al. 1996,Theorem 2.3.2). An additive mapping T : R R is called a Jordan left centralizer

ifT(x2)= T(x)xholds for all x R. In(1992), Bresar and Zalar proved that every

Jordan left centralizer on a prime ring is a left centralizer. Further, Zalar (1991) proved

this result in the setting of semiprime ring of characteristic different from two. More

related results on centralizers in rings and algebras can be looked inAli and Fosner

(2010), Bresar and Zalar(1992),Hentzel and Tammam El-Sayiad(2011), Fosner and

Vukman (2007) andZalar(1991) where further references can be found.

LetR be a ring with involution. According toAli and Fosner(2010), an additive

mappingT : R Ris said to be a left-centralizer (resp. reverse left-centralizer)

ifT(x y)= T(x)y (resp.T(x y)= T(y)x) holds for allx,y R. An additive map-

pingT : R Ris called a Jordan left -centralizer in caseT(x2)= T(x)x holds for

allx R. The definition of a right-centralizer and Jordan right-centralizer should

be self-explanatory. For some fixed element a R, the map x ax is a Jordan

left-centralizer and the map x xais a Jordan right-centralizer on R. Clearly,

every left-centralizer on a ring R is a Jordan left-centralizer. Thus, it is natural to

question that whether the converse of above statement is true. In Sect.2, it is shown

that the answer to this question is affirmative if the underlying -ringR is semiprime,

of characteristic different from two. Further, we establish a result concerning additive

mappingT : R Rsatisfying the relation T(x3)= xT(x)x for allx R.

The third section is inspired by the work of Vukman (1997, Theorem 4). He showed

that if R is a noncommutative 2-torsion free semiprime ring and S, T : R R are

left centralizers such that[S(x), T(x)]S(x) +S(x)[S(x), T(x)] =0 for allx,y R,

then[S(x), T(x)] = 0 for all x R. In case R is prime ring and S = 0(T = 0),

1 3


5/18

Beitr Algebra Geom (2013) 54:609624 611

then there exists Csuch thatT =S(S=T). The intent of Sect.3is to study

similar types of problems in the setting of ring with involution by replacing left

centralizer with Jordan left-centralizer.

We shall restrict our attention on Jordan left -centralizers, since all results pre-

sented in this article are also true for Jordan right -centralizers because of left-rightsymmetry.

2 Preliminaries

We shall do a great deal of calculations with commutators and anti-commutators and

routinely use the following basic identities: For all x,y,z R, we have

[x y,z] = x[y,z] + [x,z]y and [x,y z] = [x,y]z+ y [x,z].

Moreover

xo(yz)= (xoy)z y[x,z] = y(xz)+ [x,y]z

and

(x y)oz =(xoz)y+ x[y,z] = x(y z) [x,z]y.

The next statement is well-known and we will use it in subsequent discussions. Webegin with the following:

Lemma 2.1 (Herstein 1976, pp. 2023)Suppose that the elements ai , bi in the cen-

tral closure of a prime ring R satisfy aixbi = 0for all x R.If bi = 0for some

i , then a i s are C-independent.

Lemma 2.2 Let R be a non-commutative prime ring with involution and let T :

R R be a Jordan left-centralizer on R. If T(x) Z(R) for all x R, then

T =0.

Proof By the assumption we have[T(x),y] =0 for all x,y R. Substitutingx2for xin the above relation, we obtain

0= [T(x2),y]

= [T(x)x,y]

= [T(x),y]x +T(x)[x,y] for all x,y R.

In view of our hypothesis, the last expression yields that T(x)[x,y] = 0 for all

x, y R.Since centre of a prime ring is free from zero divisors, either T(x)=0 or

[x

,y] = 0. Let A = {x R| T(x)= 0}and B = {x R| [x

,y] =0 f or all y R}. It can be easily seen that A and B are two additive subgroups ofR whose union

is Rand hence by Brauers trick, we get A= Ror B = R. IfB = R, then Ris com-

mutative, which gives a contradiction. Thus the only possibility remains that A= R.

That is,T(x)= 0 for all x R. This finishes the proof.

1 3


6/18


The next result is motivated by the Proposition 1.4 in Zalar(1991)

Proposition 2.3 Let R be a semiprime ring with involution of characteristic dif-

ferent from two and T : R R an additive mapping which satisfies T(x2)= T(x)x

for all x R. Then T is a reverse left-centralizer that is, T(x y) = T(y)x

for allx, y R.

Proof We have T(x2)= T(x)x for allx R. Applying involution both sides to

the above expression, we obtain

(T(x2))

= x(T(x))

for allx R. Define a new map S: R Rsuch that S(x)= (T(x)) for allx R.

Then we see that

S(x2)= (T(x2))

= (T(x)x)

= x(T(x))

= x S(x)

for all x R. Hence, we obtain S(x2) = x S(x) for all x R. Thus, S is a Jordan

right-centralizer on R. In view of Proposition 1.4 inZalar(1991),Sis a right-central-

izer that is, S(x y)= x S(y)for all x,y R. This implies that (T(x y)) = x(T(y))

for allx,y R. By applying involution to the both sides of the last relation, we findthatT(x y)= T(y)x for allx,y R. This completes the proof of the proposition.

Proposition 2.4 Let R be a prime ring with involution of characteristic different

from two and T : R R an additive mapping which satisfies T(x3)= xT(x)x for

all x R. Then T(x y) = T(y)x = yT(x)for all x, y R that is, T is a reverse

-centralizer on R.

ProofBy the given hypothesis, we have T(x3)= xT(x)x for allx R. Applying

involution both sides to the above expression, we get

(T(x3))

= x(T(x))x

for allx R. Define a new map S: R Rsuch that S(x)= (T(x)) for allx R.

Then we see that

S(x3)= (T(x3))

= (xT(x)x)

= x(T(x))x

= x S(x)x

for all x R. Hence, we conclude that S(x3) = x S(x)x for all x R. Thus, S is

an additive mapping such that S(x3) = x S(x)x for all x R. In view of Fosner

and Vukman (2007, Theorem 1), we are forced to conclude that S is a two sided-

centralizer that is, S(x y) = x S(y) = S(x)y for all x,y R. This implies that

1 3


7/18


(T(x y)) = x(T(y)) = (T(x))y for all x,y R. Again applying involution both

sides to the last relation, we find that T(x y) = T(y)x = yT(x) for all x,y R.

With this the proposition is proved.

Proposition 2.5 Let R be a noncommutative prime ring with involution and letS, T : R R be Jordan left-centralizers. Suppose that[ S(x), T(x)] = 0 holds

for all x R. If T =0, then there exists C such that S=T.

Proof By Proposition2.3we conclude that Sand Tare reverse left-centralizers on

R. In view of the hypothesis, we have

[S(x), T(x)] =0 for all x R. (1)

Linearizing (1) and using it, we get

[S(x), T(y)] + [S(y), T(x)] =0 for all x, y R. (2)

Replacingxbyz x in (2), we obtain

[S(x), T(y)]z +S(x)[z, T(y)] + [S(y), T(x)]z +T(x)[S(y),z] =0 (3)

for allx,y R. Application of (2)yields that

S(x)[z, T(y)] + T(x)[S(y),z] =0 for all x,y R. (4)

Replacingxbywx in (4), we get

S(x)w[z, T(y)] + T(x)w[S(y),z] =0 for all x, y, z, w R. (5)

Replacingw by w andz byz in (5), we obtain

S(x)w[z, T(y)] + T(x)w[S(y),z] =0 for all x, y, z, w R. (6)

It follows from Lemma2.2 that there exists y,z R such that[T(y),z] = 0, since

T =0. In view of Lemma 2.1 and from relation (6)weconcludethat S(x)= (x)T(x),

where(x)is fromC. Thus the relation (6)forces that

0= (x)T(x)w[T(y),z] T(x)w[(y)T(y),z]

= (x)T(x)w[T(y),z] T(x)w(y)[T(y),z]

= ((x)(y))T(x)w[T(y),z]

for all y,z R. Since R is a prime ring, the above expression yields that either

((x) (y))T(x) = 0 or [T(y),z] = 0. Since [T(y),z] = 0, we have ((x)

(y))T(x) = 0 for all x,y R. This implies that (x)T(x) = (y)T(x) for all

x,y R. This gives S(x)= (y)T(x)for allx,y R, as desired.

1 3


8/18


If we replace the commutator by anti-commutator in Proposition 2.5, the corre-

sponding result also holds.

Proposition 2.6 Let R be a noncommutative prime ring with involution and let

S, T : R R be Jordan left-centralizers. Suppose that S(x)oT(x)= 0 holds forall x R. If T =0, then there exists C such that S=T.

ProofBy the assumption, we have

S(x) T(x)= 0 for all x R. (7)

Replacingxby x+y in (7), we obtain

S(x)T(x)+ S(x)T(y)+ S(y) T(x)+ S(y)T(y) = 0 f or all x ,y R.

(8)

Using (7) in(8), we get

S(x)T(y)+ S(y) T(x)= 0 for all x, y R. (9)

Substitutingzyforyin (9) and using the fact thatSand Tare reverse left -centralizers,we find that

0= S(x) T(zy)+ S(zy)T(x)

= S(x)(T(y)z)+T(x)(S(y)z)

= (S(x) T(y))z T(y)[S(x),z] + (T(x) S(y))z S(y)[T(x),z]

Application of (9) yields that

T(y)[S(x),z] + S(y)[T(x),z] =0 for all x,y,z R. (10)

Replacingy bywy in(10), we obtain

T(y)w[S(x),z] + S(y)w[T(x),z] =0 for all x, y, z, w R. (11)

Replacingw by w andz byz in (12), we get

T(y)w[S(x),z] + S(y)w[T(x),z] =0 for allx, y, z, w R. (12)

Henceforth using similar approach as we have used after equation (6) in the proof of

Proposition2.5,we get the required result. This finishes the proof of the proposition.

1 3


9/18


3 Main Results

The main result of the present paper is the following theorem which is inspired by

Vukmans result (Vukman 1997, Theorem 4).

Theorem 3.1 Let R be a noncommutative 2-torsion free semiprime ring with involu-

tion and S, T : R R be Jordan left-centralizers. Suppose that

(S(x) T(x))S(x) S(x)(S(x) T(x))= 0

holds for all x R. Then[ S(x), T(x)] = 0for all x R. Moreover if R is a prime

ring and S=0(T =0), then there exists C such that T = S(S=T).

ProofIn view of Proposition2.3we conclude thatSand Tare reverse left-central-

izers. By the hypothesis, we have

(S(x)T(x))S(x) S(x)(S(x)T(x))= 0 for allx R. (13)

Linearization of relation (13)yields that

(S(x)T(x))S(y)+(S(x)T(y))S(x)+(S(x) T(y))S(y)+(S(y) T(x))S(x)

+(S(y) T(x))S(y)+(S(y) T(y))S(x) S(y)(S(x) T(x))

S(x)(S(x)T(y)) S(y)(S(x)T(y)) S(x)(S(y) T(x))

S(y)(S(y)T(x)) S(x)(S(y)T(y))= 0 (14)

for allx,y R.Replacing xbyx in(14), we get

(S(x)T(x))S(y)+(S(x) T(y))S(x)(S(x) T(y))S(y)+(S(y) T(x))S(x)

(S(y) T(x))S(y)(S(y) T(y))S(x) S(y)(S(x) T(x))

S(x)(S(x)T(y))+ S(y)(S(x)T(y)) S(x)(S(y) T(x))

+S(y)(S(y)T(x))+ S(x)(S(y)T(y))= 0 (15)

for allx,y R.Combining (14) and (15), we obtain

2(S(x)T(x))S(y)+2(S(x)T(y))S(x)+2(S(y)T(x))S(x)

2S(y)(S(x) T(x))2S(x)(S(x) T(y))2S(x)(S(y) T(x))= 0

for allx,y R.Since R is 2-torsion free, the above relation reduces to

(S(x)T(x))S(y)+(S(x) T(y))S(x)+(S(y)T(x))S(x)

S(y)(S(x)T(x)) S(x)(S(x) T(y)) S(x)(S(y) T(x))= 0 (16)

for allx,y R.Replacing yby yx in (16), we obtain

(S(x) T(x))S(x)y +(S(x) T(x)y)S(x)+(S(x)y T(x))S(x)

S(x)y(S(x)T(x)) S(x)(S(x) T(x)y) S(x)(T(x) S(x)y)= 0

1 3


10/18


for allx,y R.By using anti-commutator identity, the above relation can be written

as

(S(x) T(x))S(x)y +(S(x) T(x))y S(x)T(x)[S(x),y ]S(x)

+(T(x) S(x))y S(x) S(x)[T(x),y]S(x) S(x)y(S(x)T(x))

S(x)(S(x) T(x))y +S(x)T(x)[S(x),y ] S(x)(T(x) S(x))y

+S(x)2[T(x),y] =0 (17)

for allx,y R.In view of (13), (17) reduces to

(S(x) T(x))y S(x)T(x)[S(x),y ]S(x)+(T(x) S(x))y S(x)

S(x)[T(x),y ]S(x) S(x)y(S(x) T(x)) S(x)(S(x)T(x))y

+S(x)T(x)[S(x),y] + S(x)2[T(x),y ] =0 (18)

for allx,y R.Upon substituting S(x)y for y in(18), we get

(S(x) T(x))y S(x)2 T(x)[S(x),y S(x)]S(x)+(T(x) S(x))y S(x)2 S(x)

[T(x),y S(x)]S(x) S(x)y S(x)(S(x)T(x)) S(x)(S(x) T(x))y S(x)

+S(x)T(x)[S(x),y S(x)] + S(x)2[T(x),y S(x)] =0.

This implies that

(S(x)T(x))y S(x)2 T(x)[S(x),y]S(x)2 +(T(x) S(x))y S(x)2

S(x)[T(x),y]S(x)2 S(x)y[T(x),S(x)]S(x) S(x)y S(x)(S(x) T(x))

S(x)(S(x) T(x))y S(x)+ S(x)T(x)[S(x),y]S(x)+ S(x)2[T(x),y ]S(x)

+S(x)2y[T(x),S(x)] =0 (19)

for allx,y R.Application of (18) yields that

S(x)y[T(x),S(x)]S(x) S(x)2y[T(x),S(x)] =0 (20)

for allx,y R.Replacing yby yT(x) in(20), we have

S(x)T(x)y[S(x), T(x)]S(x) S(x)2T(x)y[S(x), T(x)] =0 (21)

for allx,y R.Left multiplying (20) byT(x)gives

T(x)S(x)y[S(x), T(x)]S(x)T(x)S(x)2y[S(x), T(x)] =0

(22)

for allx,y R.On combining (21) and(22), we obtain

[S(x), T(x)]y[S(x), T(x)]S(x) [ S(x)2, T(x)]y[S(x), T(x)] =0 (23)

1 3


11/18


for allx,y R.By our hypothesis, we have

0 = (S(x)T(x))S(x) S(x)(S(x) T(x))

= S(x)T(x)S(x)+T(x)S(x)2 S(x)2T(x) S(x)T(x)S(x)

= T(x)S(x)2 S(x)2T(x)

for allx,y R.The above expression can be further written as

[S(x)2, T(x)] =0 (24)

for allx R.Using (24) in(23), we get

[S(x), T(x)]y

[S(x), T(x)]S(x)= 0 (25)

for allx,y R.Replacing yby y S(x) in (25), we obtain

[S(x), T(x)]S(x)y[S(x), T(x)]S(x)= 0 (26)

for allx,y R.Since R is a semiprime ring it follows from relation (26) that

[S(x), T(x)]S(x)= 0 (27)

for allx R.In view of relation(24) and(27), we have

S(x)[S(x), T(x)] =0 (28)

for all x,y R.Replacing x by x+ y in (28)and using the same techniques as we

used to obtain (16) from (13), we get

S(y)[S(x), T(x)] + S(x)[S(y), T(x)] + S(x)[S(x), T(y)] =0 (29)

for allx,y R.Substituting yx for y in(29), we obtain

S(x)y[S(x), T(x)] + S(x)2[y, T(x)] + S(x)[S(x), T(x)]y

+S(x)[S(x), T(x)]y +S(x)T(x)[S(x),y] =0

for allx,y R.This implies

S(x)y[S(x), T(x)] + S(x)2[y, T(x)] + S(x)T(x)[S(x),y] =0 (30)

for allx,y R.Thus we have the relation

S(x)y[S(x), T(x)] + S(x)2[y, T(x)] + S(x)T(x)[S(x),y ] =0

1 3


12/18


for allx,y R.Which can be further written in the form

S(x)y[S(x), T(x)] + S(x)2yT(x) S(x)T(x)y S(x)+ S(x)[T(x),S(x)]y =0

for allx,y R.Application of (28) forces that

S(x)y[S(x), T(x)] + S(x)2yT(x) S(x)T(x)y S(x)= 0 (31)

for allx,y R.Left multiplication of (31) byT(x)gives

T(x)S(x)y[S(x), T(x)] + T(x)S(x)2yT(x)T(x)S(x)T(x)y S(x)= 0 (32)

for allx,y R.On substituting yT(x) for y in (31), we have

S(x)T(x)y[S(x), T(x)] + S(x)2T(x)yT(x) S(x)T(x)2y S(x)= 0 (33)


[S(x), T(x)]y[S(x), T(x)]+[ S(x)2, T(x)]yT(x)+[T(x),S(x)]T(x)y S(x)= 0

(34)

for allx,y R.Using (24), the above expression reduces to

[S(x), T(x)]y[S(x), T(x)] + [T(x),S(x)]T(x)y S(x)= 0 (35)

for allx,y R.Substitutingz S(x)y for yin (35), we get

[S(x), T(x)]y S(x)z[S(x), T(x)] + [T(x),S(x)]T(x)y S(x)z S(x)= 0 (36)

for allx,y,z R.On the other hand right multiplying to (35) byz S(x), we get

[S(x), T(x)]y[S(x), T(x)]z S(x)+ [T(x),S(x)]T(x)y S(x)z S(x)= 0 (37)

for allx,y,z R.On comparing (36) and (37), we obtain

[S(x), T(x)]yA(x,z)= 0 (38)

for all x,y,z R, where A(x,z) = [S(x), T(x)]z S(x) S(x)z[S(x), T(x)].

Substitutingy S(x)zfor y in (38) gives

[S(x), T(x)]z S(x)yA(x,z)= 0 (39)

for allx,y,z R.Left multiplying to (38) by S(x)z, we get

S(x)z[S(x), T(x)]yA(x,z)= 0 (40)

1 3


13/18


for all x,y,z R. From (39) and (40), we arrive at A(x,z)y A(x,z) = 0 for all

x,y,z R. That is, A(x,z)R A(x,z) = (0)for all x,z R. The semiprimeness of

Rforces that A(x,z)= 0 for all x,z R. In other words, we have

[S(x), T(x)]z S(x)= S(x)z[S(x), T(x)] (41)

for allx,z R.Replacingz by yT(x) in (41), we have

[S(x), T(x)]T(x)y S(x)= S(x)T(x)y[S(x), T(x)] (42)


[S(x), T(x)]y[S(x), T(x)] S(x)T(x)y[S(x), T(x)] =0

for allx,y R.This further reduces to

T(x)S(x)y[S(x), T(x)] =0 (43)

for allx,y R.If we substitute yT(x) for y in (43), we find that

T(x)S(x)T(x)y[S(x), T(x)] =0 (44)

for allx,y R.Multiplying (43)from the left side by T(x), we get

T(x)2 S(x)y[S(x), T(x)] =0 (45)

for allx,y R.Subtracting(45) from (44), we get

T(x)[S(x), T(x)]y[S(x), T(x)] =0 (46)

for allx,y R.ReplacingT(x)yfor y in (46), we obtain

T(x)[S(x), T(x)]yT(x)[S(x), T(x)] =0 (47)

for allx,y R.That is,

T(x)[S(x), T(x)]RT(x)[S(x), T(x)] =(0)

for allx R. The semiprimeness ofR yields that

T(x)[S(x), T(x)] =0 (48)

for allx R.Replacing ybyT(x)y in (42)gives, because of(48)

[S(x), T(x)]yT(x)S(x)= 0 (49)

1 3


14/18


for allx,y R.Substituting x+ y for x in (27) and using the same approach as we

used to obtain (16) from (13), we get

[S(x), T(x)]S(y)+ [S(x), T(y)]S(x)+ [ S(y), T(x)]S(x)= 0 (50)

for x,y R.On substituting yx for y in (50), we obtain

[S(x), T(x)]S(x)y +T(x)[S(x),y]S(x)+ [ S(x), T(x)]y S(x)

+[S(x), T(x)]y S(x)+ S(x)[y, T(x)]S(x)= 0

for allx,y R.Application of (27) yields that

[S(x), T(x)]y S(x)+T(x)[S(x),y]S(x)+ [ S(x), T(x)]y S(x) (51)

+S(x)[y, T(x)]S(x)= 0

for allx,y R.This implies that

2[S(x), T(x)]y S(x)+T(x)[S(x),y ]S(x)+ S(x)[y, T(x)]S(x)= 0 (52)

for allx,y R.This can be further written as

2[S(x), T(x)]y

S(x)+T(x)S(x)y

S(x) T(x)y

S(x)2

+S(x)yT(x)S(x) S(x)T(x)y S(x)= 0

for allx,y R,which reduces to

[S(x), T(x)]y S(x)+ S(x)yT(x)S(x)T(x)y S(x)2 =0 (53)

for allx,y R.Using (41) in(53), we obtain

0= S(x)y[S(x), T(x)] + S(x)yT(x)S(x)T(x)y S(x)2

= S(x)y S(x)T(x) T(x)y S(x)2

for allx,y R.The above expression yields that

S(x)y S(x)T(x)= T(x)y S(x)2 (54)

for allx,y R.Substituting yT(x) for y in (54), we have

S(x)T(x)y S(x)T(x)= T(x)2y S(x)2 (55)

for allx,y R.Left multiplication to(54) by T(x)leads to

T(x)S(x)y S(x)T(x)= T(x)2y S(x)2 (56)

1 3


15/18


for allx,y R.By combining (55) and (56), we arrive at

[S(x), T(x)]y S(x)T(x)= 0 (57)

for allx,y R.From (49)and (57), we obtain

[S(x), T(x)]y[S(x), T(x)] =0

for allx,y R.That is,[S(x), T(x)]R[S(x), T(x)] =(0).The semiprimeness ofR

yields that[S(x), T(x)] =0 for allx R.IfR is prime, then in view of Proposition

2.5 we get the required result. Thereby the proof of theorem is completed.

Theorem 3.2 Let R be a noncommutative 2-torsion free semiprime ring with involu-

tion and S, T : R R be Jordan left-centralizers. Suppose that

[S(x), T(x)]S(x) S(x)[S(x), T(x)] =0

holds for all x R. Then[ S(x), T(x)] = 0for all x R. Moreover if R is a prime

ring and S=0(T =0), then there exists C such that T = S(S=T).

Proof We notice that S and T are reverse left -centralizers by Proposition2.3. By

the assumption we have the relation

[S(x), T(x)]S(x) S(x)[S(x), T(x)] =0 (58)

for allx R.Replacing x by x+ y in(58) and using similar techniques as we used

to obtain(16) from (13), we find that

[S(x), T(x)]S(y)+ [ S(x), T(y)]S(x)+ [S(y), T(x)]S(x)

S(y)[S(x), T(x)] S(x)[S(x), T(y)] S(x)[S(y), T(x)] =0 (59)

for allx,y R.Substituting yx for y in(59), we obtain

[S(x), T(x)]S(x)y + [S(x), T(x)]y S(x)+T(x)[S(x),y ]S(x)

+[S(x), T(x)]y S(x)+ S(x)[y, T(x)]S(x) S(x)y[S(x), T(x)]

S(x)T(x)[S(x),y] S(x)[S(x), T(x)]y

S(x)2[y, T(x)] S(x)[S(x), T(x)]y =0 (60)

for allx,y R.Application of (58) forces that

2[S(x), T(x)]y S(x)+T(x)[S(x),y]S(x)+ S(x)[y, T(x)]S(x)

S(x)y[S(x), T(x)] S(x)T(x)[S(x),y]

S(x)[S(x), T(x)]y S(x)2[y, T(x)] =0 (61)

1 3


16/18


for allx,y R.Substituting S(x)yfor y in(61), we have

2[S(x), T(x)]y S(x)2 +T(x)[S(x),y ]S(x)2 +S(x)[y, T(x)]S(x)2

+S(x)y[S(x), T(x)]S(x) S(x)y S(x)[S(x), T(x)] S(x)T(x)[S(x),y ]S(x)

S(x)[S(x), T(x)]y S(x) S(x)2y[S(x), T(x)] S(x)2[y, T(x)]S(x)= 0

(62)

for allx,y R.Using (61) in(62), we conclude that

S(x)y[S(x), T(x)]S(x) S(x)2y[S(x), T(x)] =0 (63)

for allx,y R.Substituting yT(x) for y in the above relation, we obtain

S(x)T(x)y[S(x), T(x)]S(x) S(x)2T(x)y[S(x), T(x)] =0 (64)

for allx,y R.On the other hand left multiplication of (63) byT(x)gives

T(x)S(x)y[S(x), T(x)]S(x)T(x)S(x)2y[S(x), T(x)] =0 (65)

for allx,y R.By comparing (64) and (65), we obtain

0= [S(x), T(x)]y[S(x), T(x)]S(x) [ S(x)2, T(x)]y[S(x), T(x)]

= [S(x), T(x)]y[S(x), T(x)]S(x)([S(x), T(x)]S(x)

+S(x)[S(x), T(x)])y[S(x), T(x)]

for allx,y R.In view of the hypothesis, the above expression reduces to

[S(x), T(x)]y[S(x), T(x)]S(x)2S(x)[S(x), T(x)]y[S(x), T(x)] (66)

for allx,y R.If we multiply(66) by S(x)from left, we get

S(x)[S(x), T(x)]y[S(x), T(x)]S(x)2S(x)2[S(x), T(x)]y[S(x), T(x)] =0

(67)

for allx,y R.On the other hand putting y[S(x), T(x)] for y in(63), we arrive at

S(x)[S(x), T(x)]y[S(x), T(x)]S(x) S(x)2[S(x), T(x)]y[S(x), T(x)] =0

(68)

for allx,y R.By combining (67) and (68), we obtain

S(x)[S(x), T(x)]y[S(x), T(x)]S(x)= 0 for all x,y R.

1 3


17/18


Using (58)in the above expression, we obtain

S(x)[S(x), T(x)]y S(x)[S(x), T(x)] =0 for all x,y R.

Since R is semiprime, it follows that

S(x)[S(x), T(x)] =0 (69)

for allx R.From(69) and(58), we get

[S(x), T(x)]S(x)= 0 (70)

for all x R. The last two expressions are same as the equations (27) &(28)and

hence, by using similar approach as we have used after (27) &(28) in the proof ofTheorem3.1,we get the required result. The theorem is thereby proved.

The following results are immediate consequences of the above theorems.

Corollary 3.3 Let R be a noncommutative 2-torsion free semiprime ring with invo-

lution and T : R R a Jordan left-centralizer. Suppose (T(x) x)x

x(T(x) x) = 0 holds for all x R. Then T is a reverse -centralizer

on R.

Proof Taking S(x) = x

in Theorem3.1 and using the fact that the product

iscommutative, we find that

[T(x),x] =0

for allx R. From the above relation, we obtain T(x2)= T(x)x = xT(x)for all

x R.This shows thatTis Jordan left as well as right-centralizer on R. Hence by

Proposition2.3we conclude thatTis a reverse-centralizer on R.

Similarly, we prove the following:

Corollary 3.4 Let R be a noncommutative 2-torsion free semiprime ring with invo-

lution and T : R R a Jordan left -centralizer. Suppose (T(x) x)T(x)

T(x)(T(x)x)= 0 holds for all x R. Then, T is a reverse -centralizer on R.

Corollary 3.5 Let R be a noncommutative 2-torsion free semiprime with involution

and T : R R a Jordan left-centralizer. Suppose [T(x),x]x x[T(x),x] =

0holds for all x R. In this case, T is a reverse -centralizer on R.

Proof Substituting S(x)= x in Theorem3.2,we obtain

[T(x),x] =0

for allx R.This implies thatT(x2)= T(x)x = xT(x)for allx R.In view of

Proposition 2.3, we conclude thatTis a reverse-centralizer on R.

1 3


18/18


Corollary 3.6 Let R be a noncommutative 2-torsion free semiprime with involu-

tion and T : R R a Jordan left -centralizer. Suppose [T(x),x]T(x)

T(x)[T(x),x] = 0 holds for all x R. In this case, T is a reverse-centralizer

on R.

Acknowledgments The authors are greatly indebted to the referee for his\her valuable comments and

suggestions. We take this opportunity to express our gratitude to Professor Daniel Eremita for his useful

discussions.

References

Ali, S., Fosner, A.: On Jordan(, )-derivations in semiprime-rings. Int. J. Algebra4(3), 99108 (2010)

Beidar, K.I., Martindale III, W.S., Mikhalev, A.V.: Rings with generalized identities. Marcel Dekker Inc.,

New York (1996)

Bresar, M., Zalar, B.: On the structure of Jordan-derivations, Colloq. Mathematics63(2), 163171 (1992)Fosner, M., Vukman, J.: A characterization of two sided centralizers on prime ring. Taiwan J. Math 11,

14311441 (2007)

Hentzel, I.R., Tammam El-Sayiad, M.S.: Left centralizers on rings that are not semiprime. Rocky Mountain

J. Math.41(5):14711481 (2011)

Herstein I.N. (1976) Ring with involution, University of Chicago Press, Chicago

Martindale, III. W.S.: Prime rings satisfying a generalized polynomial identity. J. Algebra 12, 576584

(1969)

Vukman, J.: Centralizers on prime and semiprime rings. Comment Math. Univ. Carolin 38(2), 231240

(1997)

Vukman, J.: Centralizers on semiprime rings. Comment Math. Univ. Carolin 42, 245273 (2001)

Zalar, B.: On centralizers of semiprime rings. Comment. Math. Univ. Carolin 32(4), 609614 (1991)

1 3

Documents

Reprints From BAG2013