Jordan left *-centralizers of prime and
semiprime rings with involution
Shakir Ali, Nadeem Ahmad Dar & Joso
Vukman
Beiträge 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
1 23
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 your
article, 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”.
1 23
Author's personal copy
Beitr Algebra Geom (2013) 54:609–624
DOI 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) if T (x y) = T (x)y ∗ (resp.
T (x 2 ) = T (x)x ∗ ) holds for all x, y ∈ R, and a reverse left ∗-centralizer if T (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 related
results 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: shakir50@rediffmail.com; shakir.ali.mm@amu.ac.in
N. A. Dar
e-mail: ndmdarlajurah@gmail.com
J. Vukman
Department of Mathematics and Computer Science,
University of Maribor FNM, Koroska 160, 2000 Maribor, Slovenia
e-mail: joso.vukman@uni-mb.si
123
Author's personal copy
610
Beitr Algebra Geom (2013) 54:609–624
1 Introduction
This paper deals with the study of Jordan left ∗-centralizers of prime and semiprime rings with involution ∗ , and was motivated by the work of Vukman (1997)
and Zalar (1991). Throughout, R will represent an associative ring with centre Z (R).
We shall denote by C the extended centroid of a prime ring R. For the explanation of
C we refer the reader to Martindale (1969). Given an integer n ≥ 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 − yx and anti-commutator x y + yx, respectively. An
additive mapping x → x ∗ on a ring R is said to be an involution if (x y)∗ = y ∗ x ∗
and (x ∗ )∗ = x holds for x, 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 case
T (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 x for all x ∈ R, where q is an element of Martindale right
ring of quotients Q r of R (see Beidar et al. 1996, Chapter 2 for details). In case R has
an identity element, then T : R → R is a left centralizer if and only if T is 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 T is called a two-sided
centralizer in case T : R → R is a left and a right centralizer. In case T : R → R is a
two-sided centralizer, where R is a semiprime ring with extended centoid C, then there
exists an element λ ∈ C such that T (x) = λx for all x ∈ R (viz; Beidar et al. 1996,
Theorem 2.3.2). An additive mapping T : R → R is called a Jordan left centralizer
if T (x 2 ) = T (x)x holds for all x ∈ R. In (1992), Bres̆ar 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 in Ali and Fos̆ner
(2010), Bres̆ar and Zalar (1992), Hentzel and Tammam El-Sayiad (2011), Fos̆ner and
Vukman (2007) and Zalar (1991) where further references can be found.
Let R be a ring with involution ∗ . According to Ali and Fos̆ner (2010), an additive
mapping T : R → R is said to be a left ∗-centralizer (resp. reverse left ∗-centralizer)
if T (x y) = T (x)y ∗ (resp. T (x y) = T (y)x ∗ ) holds for all x, y ∈ R. An additive mapping T : R → R is called a Jordan left ∗-centralizer in case T (x 2 ) = T (x)x ∗ holds for
all x ∈ 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 → x ∗ a is 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 ∗ -ring R is semiprime,
of characteristic different from two. Further, we establish a result concerning additive
mapping T : R → R satisfying the relation T (x 3 ) = x ∗ T (x)x ∗ for all x ∈ 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 all x, y ∈ R,
then [S(x), T (x)] = 0 for all x ∈ R. In case R is prime ring and S = 0(T = 0),
123
Author's personal copy
Beitr Algebra Geom (2013) 54:609–624
611
then there exists λ ∈ C such that T = λS (S = λT ). The intent of Sect. 3 is 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 presented in this article are also true for Jordan right ∗-centralizers because of left-right
symmetry.
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, yz] = [x, y]z + y[x, z].
Moreover
xo(yz) = (xoy)z − y[x, z] = y(x ◦ z) + [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. We
begin with the following:
Lemma 2.1 (Herstein 1976, pp. 20–23) Suppose that the elements ai , bi in the central closure of a prime ring R satisfy ai xbi = 0 for all x ∈ R. If bi = 0 for some
i, then ai 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 x in the above relation, we obtain
0 = [T (x 2 ), y]
= [T (x)x ∗ , y]
= [T (x), y]x ∗ + T (x)[x ∗ , y]
for all
for all
x, y ∈ R. Substituting x 2
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 of R whose union
is R and hence by Brauer’s trick, we get A = R or B = R. If B = R, then R is commutative, 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.
123
Author's personal copy
612
Beitr Algebra Geom (2013) 54:609–624
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 different from two and T : R → R an additive mapping which satisfies T (x 2 ) = T (x)x ∗
for all x ∈ R. Then T is a reverse left ∗-centralizer that is, T (x y) = T (y)x ∗ for all
x, y ∈ R.
Proof We have T (x 2 ) = T (x)x ∗ for all x ∈ R. Applying involution ∗ both sides to
the above expression, we obtain
∗
(T (x 2 )) = x(T (x))∗
for all x ∈ R. Define a new map S : R → R such that S(x) = (T (x))∗ for all x ∈ R.
Then we see that
S(x 2 ) = (T (x 2 ))∗
= (T (x)x ∗ )∗
= x(T (x))∗
= x S(x)
for all x ∈ R. Hence, we obtain S(x 2 ) = x S(x) for all x ∈ R. Thus, S is a Jordan
right-centralizer on R. In view of Proposition 1.4 in Zalar (1991), S is a right-centralizer that is, S(x y) = x S(y) for all x, y ∈ R. This implies that (T (x y))∗ = x(T (y))∗
for all x, y ∈ R. By applying involution to the both sides of the last relation, we find
that T (x y) = T (y)x ∗ for all x, 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 (x 3 ) = x ∗ T (x)x ∗ for
all x ∈ R. Then T (x y) = T (y)x ∗ = y ∗ T (x) for all x, y ∈ R that is, T is a reverse
∗-centralizer on R.
Proof By the given hypothesis, we have T (x 3 ) = x ∗ T (x)x ∗ for all x ∈ R. Applying
involution ∗ both sides to the above expression, we get
∗
(T (x 3 )) = x(T (x))∗ x
for all x ∈ R. Define a new map S : R → R such that S(x) = (T (x))∗ for all x ∈ R.
Then we see that
S(x 3 ) = (T (x 3 ))∗
= (x ∗ T (x)x ∗ )∗
= x(T (x))∗ x
= x S(x)x
for all x ∈ R. Hence, we conclude that S(x 3 ) = x S(x)x for all x ∈ R. Thus, S is
an additive mapping such that S(x 3 ) = x S(x)x for all x ∈ R. In view of Fos̆ner
and Vukman (2007, Theorem 1), we are forced to conclude that S is a two sidedcentralizer that is, S(x y) = x S(y) = S(x)y for all x, y ∈ R. This implies that
123
Author's personal copy
Beitr Algebra Geom (2013) 54:609–624
613
(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 ∗ = y ∗ T (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 let
S, 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 λ ∈ C such that S = λT.
Proof By Proposition 2.3 we conclude that S and T are 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)
Replacing x by zx 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 all x, y ∈ R. Application of (2) yields that
S(x)[z ∗ , T (y)] + T (x)[S(y), z ∗ ] = 0 for all x, y ∈ R.
(4)
Replacing x by wx 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)
Replacing w by w ∗ and z by z ∗ 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 Lemma 2.2 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) we conclude that S(x) = λ(x)T (x),
where λ(x) is from C. Thus the relation (6) forces that
0 = λ(x)T (x)w[T (y), z] − T (x)w[λ(y)T (y), z]
= λ(x)T (x)w[T (y), z] − T (x)wλ(y)[T (y), z]
= (λ(x) − λ(y))T (x)w[T (y), z]
for all y, z ∈ R. Since R is a prime ring, the above expression yields that either
(λ(x) − λ(y))T (x) = 0 or [T (y), z] = 0. Since [T (y), z] = 0, we have (λ(x) −
λ(y))T (x) = 0 for all x, y ∈ R. This implies that λ(x)T (x) = λ(y)T (x) for all
x, y ∈ R. This gives S(x) = λ(y)T (x) for all x, y ∈ R, as desired.
123
Author's personal copy
614
Beitr Algebra Geom (2013) 54:609–624
If we replace the commutator by anti-commutator in Proposition 2.5, the corresponding 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 for
all x ∈ R. If T = 0, then there exists λ ∈ C such that S = λT.
Proof By the assumption, we have
S(x) ◦ T (x) = 0 for all x ∈ R.
(7)
Replacing x by 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)
Substituting zy for y in (9) and using the fact that S and T are 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)
Replacing y by wy 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)
Replacing w by w ∗ and z by z ∗ in (12), we get
T (y)w[S(x), z] + S(y)w[T (x), z] = 0
for all x, y, z, w ∈ R.
(12)
Henceforth using similar approach as we have used after equation (6) in the proof of
Proposition 2.5, we get the required result. This finishes the proof of the proposition.
123
Author's personal copy
Beitr Algebra Geom (2013) 54:609–624
615
3 Main Results
The main result of the present paper is the following theorem which is inspired by
Vukman’s result (Vukman 1997, Theorem 4).
Theorem 3.1 Let R be a noncommutative 2-torsion free semiprime ring with involution ∗ 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)] = 0 for all x ∈ R. Moreover if R is a prime
ring and S = 0(T = 0), then there exists λ ∈ C such that T = λS(S = λT ).
Proof In view of Proposition 2.3 we conclude that S and T are reverse left ∗-centralizers. By the hypothesis, we have
(S(x) ◦ T (x))S(x) − S(x)(S(x) ◦ T (x)) = 0
for all x ∈ 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 all x, y ∈ R. Replacing x by −x 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 all x, 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 all x, 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 all x, y ∈ R. Replacing y by 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
123
Author's personal copy
616
Beitr Algebra Geom (2013) 54:609–624
for all x, 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 all x, 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 all x, 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)2 y ∗ [T (x), S(x)] = 0
(19)
for all x, y ∈ R. Application of (18) yields that
S(x)y ∗ [T (x), S(x)]S(x) − S(x)2 y ∗ [T (x), S(x)] = 0
(20)
for all x, y ∈ R. Replacing y by yT (x)∗ in (20), we have
S(x)T (x)y ∗ [S(x), T (x)]S(x) − S(x)2 T (x)y ∗ [S(x), T (x)] = 0
(21)
for all x, y ∈ R. Left multiplying (20) by T (x) gives
T (x)S(x)y ∗ [S(x), T (x)]S(x) − T (x)S(x)2 y ∗ [S(x), T (x)] = 0
(22)
for all x, 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
123
(23)
Author's personal copy
Beitr Algebra Geom (2013) 54:609–624
617
for all x, 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)2 T (x) − S(x)T (x)S(x)
= T (x)S(x)2 − S(x)2 T (x)
for all x, y ∈ R. The above expression can be further written as
[S(x)2 , T (x)] = 0
(24)
for all x ∈ R. Using (24) in (23), we get
[S(x), T (x)]y ∗ [S(x), T (x)]S(x) = 0
(25)
for all x, y ∈ R. Replacing y by y S(x)∗ in (25), we obtain
[S(x), T (x)]S(x)y ∗ [S(x), T (x)]S(x) = 0
(26)
for all x, y ∈ R. Since R is a semiprime ring it follows from relation (26) that
[S(x), T (x)]S(x) = 0
(27)
for all x ∈ 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 all x, 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 all x, 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 all x, 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
123
Author's personal copy
618
Beitr Algebra Geom (2013) 54:609–624
for all x, y ∈ R. Which can be further written in the form
S(x)y ∗ [S(x), T (x)] + S(x)2 y ∗ T (x) − S(x)T (x)y ∗ S(x) + S(x)[T (x), S(x)]y ∗ = 0
for all x, y ∈ R. Application of (28) forces that
S(x)y ∗ [S(x), T (x)] + S(x)2 y ∗ T (x) − S(x)T (x)y ∗ S(x) = 0
(31)
for all x, y ∈ R. Left multiplication of (31) by T (x) gives
T (x)S(x)y ∗ [S(x), T (x)] + T (x)S(x)2 y ∗ T (x) − T (x)S(x)T (x)y ∗ S(x) = 0
(32)
for all x, y ∈ R. On substituting yT (x)∗ for y in (31), we have
S(x)T (x)y ∗ [S(x), T (x)] + S(x)2 T (x)y ∗ T (x) − S(x)T (x)2 y ∗ S(x) = 0
(33)
for all x, y ∈ R. Combining (32) and (33), we obtain
[S(x), T (x)]y ∗ [S(x), T (x)]+[S(x)2 , T (x)]y ∗ T (x)+[T (x), S(x)]T (x)y ∗ S(x) = 0
(34)
for all x, 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 all x, y ∈ R. Substituting zS(x)∗ y for y in (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 all x, y, z ∈ R. On the other hand right multiplying to (35) by z ∗ 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 all x, y, z ∈ R. On comparing (36) and (37), we obtain
[S(x), T (x)]y ∗ A(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)].
Substituting y S(x)∗ z for y in (38) gives
[S(x), T (x)]z ∗ S(x)y ∗ A(x, z) = 0
(39)
for all x, y, z ∈ R. Left multiplying to (38) by S(x)z ∗ , we get
S(x)z ∗ [S(x), T (x)]y ∗ A(x, z) = 0
123
(40)
Author's personal copy
Beitr Algebra Geom (2013) 54:609–624
619
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
R forces 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 all x, z ∈ R. Replacing z 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 all x, 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 all x, y ∈ R. This further reduces to
T (x)S(x)y ∗ [S(x), T (x)] = 0
(43)
for all x, 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 all x, 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 all x, y ∈ R. Subtracting (45) from (44), we get
T (x)[S(x), T (x)]y ∗ [S(x), T (x)] = 0
(46)
for all x, y ∈ R. Replacing T (x)∗ y for y in (46), we obtain
T (x)[S(x), T (x)]y ∗ T (x)[S(x), T (x)] = 0
(47)
for all x, y ∈ R. That is,
T (x)[S(x), T (x)]RT (x)[S(x), T (x)] = (0)
for all x ∈ R. The semiprimeness of R yields that
T (x)[S(x), T (x)] = 0
(48)
for all x ∈ R. Replacing y by T (x)∗ y in (42) gives, because of (48)
[S(x), T (x)]y ∗ T (x)S(x) = 0
(49)
123
Author's personal copy
620
Beitr Algebra Geom (2013) 54:609–624
for all x, 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 all x, 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 all x, 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 all x, 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)y ∗ T (x)S(x) − S(x)T (x)y ∗ S(x) = 0
for all x, y ∈ R, which reduces to
[S(x), T (x)]y ∗ S(x) + S(x)y ∗ T (x)S(x) − T (x)y ∗ S(x)2 = 0
(53)
for all x, y ∈ R. Using (41) in (53), we obtain
0 = S(x)y ∗ [S(x), T (x)] + S(x)y ∗ T (x)S(x) − T (x)y ∗ S(x)2
= S(x)y ∗ S(x)T (x) − T (x)y ∗ S(x)2
for all x, y ∈ R. The above expression yields that
S(x)y ∗ S(x)T (x) = T (x)y ∗ S(x)2
(54)
for all x, y ∈ R. Substituting yT (x)∗ for y in (54), we have
S(x)T (x)y ∗ S(x)T (x) = T (x)2 y ∗ S(x)2
(55)
for all x, y ∈ R. Left multiplication to (54) by T (x) leads to
T (x)S(x)y ∗ S(x)T (x) = T (x)2 y ∗ S(x)2
123
(56)
Author's personal copy
Beitr Algebra Geom (2013) 54:609–624
621
for all x, y ∈ R. By combining (55) and (56), we arrive at
[S(x), T (x)]y ∗ S(x)T (x) = 0
(57)
for all x, y ∈ R. From (49) and (57), we obtain
[S(x), T (x)]y ∗ [S(x), T (x)] = 0
for all x, y ∈ R. That is, [S(x), T (x)]R[S(x), T (x)] = (0). The semiprimeness of R
yields that [S(x), T (x)] = 0 for all x ∈ R. If R 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 involution ∗ 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)] = 0 for all x ∈ R. Moreover if R is a prime
ring and S = 0(T = 0), then there exists λ ∈ C such that T = λS(S = λT ).
Proof We notice that S and T are reverse left ∗-centralizers by Proposition 2.3. By
the assumption we have the relation
[S(x), T (x)]S(x) − S(x)[S(x), T (x)] = 0
(58)
for all x ∈ 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 all x, 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 all x, 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)
123
Author's personal copy
622
Beitr Algebra Geom (2013) 54:609–624
for all x, y ∈ R. Substituting S(x)∗ y for 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)2 y ∗ [S(x), T (x)] − S(x)2 [y ∗ , T (x)]S(x) = 0
(62)
for all x, y ∈ R. Using (61) in (62), we conclude that
S(x)y ∗ [S(x), T (x)]S(x) − S(x)2 y ∗ [S(x), T (x)] = 0
(63)
for all x, 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)2 T (x)y ∗ [S(x), T (x)] = 0
(64)
for all x, y ∈ R. On the other hand left multiplication of (63) by T (x) gives
T (x)S(x)y ∗ [S(x), T (x)]S(x) − T (x)S(x)2 y ∗ [S(x), T (x)] = 0
(65)
for all x, 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 all x, 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 all x, 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 all x, 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 all x, y ∈ R. By combining (67) and (68), we obtain
S(x)[S(x), T (x)]y ∗ [S(x), T (x)]S(x) = 0
123
for all x, y ∈ R.
Author's personal copy
Beitr Algebra Geom (2013) 54:609–624
623
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 all x ∈ 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 of
Theorem 3.1, 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 involution ∗ 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 Theorem 3.1 and using the fact that the product ◦ is
commutative, we find that
[T (x), x ∗ ] = 0
for all x ∈ R. From the above relation, we obtain T (x 2 ) = T (x)x ∗ = x ∗ T (x) for all
x ∈ R. This shows that T is Jordan left as well as right ∗-centralizer on R. Hence by
Proposition 2.3 we conclude that T is a reverse ∗-centralizer on R.
Similarly, we prove the following:
Corollary 3.4 Let R be a noncommutative 2-torsion free semiprime ring with involution ∗ 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 ∗ ] =
0 holds for all x ∈ R. In this case, T is a reverse ∗-centralizer on R.
Proof Substituting S(x) = x ∗ in Theorem 3.2, we obtain
[T (x), x ∗ ] = 0
for all x ∈ R. This implies that T (x 2 ) = T (x)x ∗ = x ∗ T (x) for all x ∈ R. In view of
Proposition 2.3, we conclude that T is a reverse ∗-centralizer on R.
123
Author's personal copy
624
Beitr Algebra Geom (2013) 54:609–624
Corollary 3.6 Let R be a noncommutative 2-torsion free semiprime with involution ∗ 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., Fos̆ner, A.: On Jordan (α, β)∗ -derivations in semiprime ∗-rings. Int. J. Algebra 4(3), 99–108 (2010)
Beidar, K.I., Martindale III, W.S., Mikhalev, A.V.: Rings with generalized identities. Marcel Dekker Inc.,
New York (1996)
Bres̆ar, M., Zalar, B.: On the structure of Jordan ∗-derivations, Colloq. Mathematics 63(2), 163–171 (1992)
Fos̆ner, M., Vukman, J.: A characterization of two sided centralizers on prime ring. Taiwan J. Math 11,
1431–1441 (2007)
Hentzel, I.R., Tammam El-Sayiad, M.S.: Left centralizers on rings that are not semiprime. Rocky Mountain
J. Math. 41(5):1471–1481 (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, 576–584
(1969)
Vukman, J.: Centralizers on prime and semiprime rings. Comment Math. Univ. Carolin 38(2), 231–240
(1997)
Vukman, J.: Centralizers on semiprime rings. Comment Math. Univ. Carolin 42, 245–273 (2001)
Zalar, B.: On centralizers of semiprime rings. Comment. Math. Univ. Carolin 32(4), 609–614 (1991)
123
View publication stats