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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
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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]
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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),
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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.
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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
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(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.
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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.
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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
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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)
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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
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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)
for allx,y R.Combining (32) and (33), we obtain
[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)
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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)
for allx,y R.Combining (35) and (42), we obtain
[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)
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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)
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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)
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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.
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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.
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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.
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