ORIGINAL PAPERwith involution

1 23

Beiträge zur Algebra und Geometrie /Contributions to Algebra andGeometryContributions to Algebra and Geometry ISSN 0138-4821Volume 54Number 2 Beitr Algebra Geom (2013) 54:609-624DOI 10.1007/s13366-012-0117-3

Jordan left *-centralizers of prime andsemiprime rings with involution

Shakir Ali, Nadeem Ahmad Dar & JosoVukman

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”.

Beitr Algebra Geom (2013) 54:609–624DOI 10.1007/s13366-012-0117-3

ORIGINAL PAPER

Jordan left ∗-centralizers of prime and semiprime ringswith 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 iscalled a left ∗-centralizer (resp. Jordan left ∗-centralizer) if T (xy) = T (x)y∗ (resp.T (x2) = T (x)x∗) holds for all x, y ∈ R, and a reverse left ∗-centralizer if T (xy) =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 twois a reverse left ∗-centralizer. This result makes it possible to solve some functionalequations 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. DarDepartment of Mathematics, Aligarh Muslim University, Aligarh 202002, Indiae-mail: [email protected]; [email protected]

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

J. VukmanDepartment of Mathematics and Computer Science,University of Maribor FNM, Koroska 160, 2000 Maribor, Sloveniae-mail: [email protected]

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 semi-prime 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 ofC we refer the reader to Martindale (1969). Given an integer n ≥ 2, a ring R is saidto be n-torsion free, if for x ∈ R, nx = 0 implies x = 0. As usual [x, y] and x ◦ ywill denote the commutator xy − yx and anti-commutator xy + yx , respectively. Anadditive mapping x �→ x∗ on a ring R is said to be an involution if (xy)∗ = y∗x∗and (x∗)∗ = x holds for x, y ∈ R. A ring equipped with an involution is called a ringwith involution or ∗-ring. Recall that R is prime if for a, b ∈ R, a Rb = (0) impliesa = 0 or b = 0, and is semiprime in case a Ra = (0) implies a = 0. FollowingZalar (1991), an additive mapping T : R → R is called a left centralizer in caseT (xy) = T (x)y holds for all x, y ∈ R. For a semiprime ring R, all left centralizersare of the form T (x) = qx for all x ∈ R, where q is an element of Martindale rightring of quotients Qr of R (see Beidar et al. 1996, Chapter 2 for details). In case R hasan identity element, then T : R → R is a left centralizer if and only if T is of the formT (x) = ax for all x ∈ R and some fixed element a ∈ R. The definition of a rightcentralizer should be self-explanatory. An additive mapping T is called a two-sidedcentralizer in case T : R → R is a left and a right centralizer. In case T : R → R is atwo-sided centralizer, where R is a semiprime ring with extended centoid C , then thereexists 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 centralizerif T (x2) = T (x)x holds for all x ∈ R. In (1992), Bres̆ar and Zalar proved that everyJordan left centralizer on a prime ring is a left centralizer. Further, Zalar (1991) provedthis result in the setting of semiprime ring of characteristic different from two. Morerelated 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 andVukman (2007) and Zalar (1991) where further references can be found.

Let R be a ring with involution ′∗′. According to Ali and Fos̆ner (2010), an additivemapping T : R → R is said to be a left ∗-centralizer (resp. reverse left ∗-centralizer)if T (xy) = T (x)y∗ (resp. T (xy) = T (y)x∗) holds for all x, y ∈ R. An additive map-ping T : R → R is called a Jordan left ∗-centralizer in case T (x2) = T (x)x∗ holds forall x ∈ R. The definition of a right ∗-centralizer and Jordan right ∗-centralizer shouldbe self-explanatory. For some fixed element a ∈ R, the map x �→ ax∗ is a Jordanleft ∗-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 toquestion that whether the converse of above statement is true. In Sect. 2, it is shownthat 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 additivemapping T : R → R satisfying the relation T (x3) = x∗T (x)x∗ for all x ∈ R.

The third section is inspired by the work of Vukman (1997, Theorem 4). He showedthat if R is a noncommutative 2-torsion free semiprime ring and S, T : R → R areleft 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


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 studysimilar types of problems in the setting of ring with involution ′∗′ by replacing leftcentralizer 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 androutinely use the following basic identities: For all x, y, z ∈ R, we have

[xy, 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

(xy)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. 20–23) Suppose that the elements ai , bi in the cen-tral closure of a prime ring R satisfy �ai xbi = 0 for all x ∈ R. If bi = 0 for somei , 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, thenT = 0.

Proof By the assumption we have [T (x), y] = 0 for all x, y ∈ R. Substituting x2

for x in 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 allx, 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 unionis R and hence by Brauer’s trick, we get A = R or B = R. If B = R, then R is 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. �

123



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 (xy) = T (y)x∗ for allx, y ∈ R.

Proof We have T (x2) = T (x)x∗ for all x ∈ R. Applying involution ′∗′ both sides tothe above expression, we obtain

(T (x2))∗ = 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(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 Jordanright-centralizer on R. In view of Proposition 1.4 in Zalar (1991), S is a right-central-izer that is, S(xy) = x S(y) for all x, y ∈ R. This implies that (T (xy))∗ = x(T (y))∗for all x, y ∈ R. By applying involution to the both sides of the last relation, we findthat T (xy) = 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 differentfrom two and T : R → R an additive mapping which satisfies T (x3) = x∗T (x)x∗ forall x ∈ R. Then T (xy) = 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 (x3) = x∗T (x)x∗ for all x ∈ R. Applyinginvolution ′∗′ both sides to the above expression, we get

(T (x3))∗ = 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(x3) = (T (x3))∗= (x∗T (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 isan additive mapping such that S(x3) = x S(x)x for all x ∈ R. In view of Fos̆nerand Vukman (2007, Theorem 1), we are forced to conclude that S is a two sided-centralizer that is, S(xy) = x S(y) = S(x)y for all x, y ∈ R. This implies that

123



(T (xy))∗ = x(T (y))∗ = (T (x))∗y for all x, y ∈ R. Again applying involution bothsides to the last relation, we find that T (xy) = 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 letS, T : R −→ R be Jordan left ∗-centralizers. Suppose that [S(x), T (x)] = 0 holdsfor 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 onR. 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, sinceT = 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 allx, y ∈ R. This gives S(x) = λ(y)T (x) for all x, y ∈ R, as desired. �

123



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 letS, T : R −→ R be Jordan left ∗-centralizers. Suppose that S(x)oT (x) = 0 holds forall 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 ofProposition 2.5, we get the required result. This finishes the proof of the proposition.

�

123



3 Main Results

The main result of the present paper is the following theorem which is inspired byVukman’s 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)] = 0 for all x ∈ R. Moreover if R is a primering 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 ∗-central-izers. 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



for all x, y ∈ R. By using anti-commutator identity, the above relation can be writtenas

(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)


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)2T (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 (23)

123



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)2T (x) − S(x)T (x)S(x)

= T (x)S(x)2 − S(x)2T (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 yS(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 weused 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



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)2T (x)y∗T (x) − S(x)T (x)2 y∗S(x) = 0 (33)


[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 yS(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 (40)

123



for all x, y, z ∈ R. From (39) and (40), we arrive at A(x, z)y A(x, z) = 0 for allx, y, z ∈ R. That is, A(x, z)R A(x, z) = (0) for all x, z ∈ R. The semiprimeness ofR 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)


[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)2S(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



for all x, y ∈ R. Substituting x + y for x in (27) and using the same approach as weused 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


[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 (56)

123



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 Ryields that [S(x), T (x)] = 0 for all x ∈ R. If R is prime, then in view of Proposition2.5 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)] = 0 for all x ∈ R. Moreover if R is a primering 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. Bythe 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 usedto 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



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)2T (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 for all x, y ∈ R.

123



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) andhence, by using similar approach as we have used after (27) & (28) in the proof ofTheorem 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 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 ∗-centralizeron R.

Proof Taking S(x) = x∗ in Theorem 3.1 and using the fact that the product ′◦′ iscommutative, we find that

[T (x), x∗] = 0

for all x ∈ R. From the above relation, we obtain T (x2) = T (x)x∗ = x∗T (x) for allx ∈ R. This shows that T is Jordan left as well as right ∗-centralizer on R. Hence byProposition 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 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∗] =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 (x2) = T (x)x∗ = x∗T (x) for all x ∈ R. In view ofProposition 2.3, we conclude that T is a reverse ∗-centralizer on R. �

123



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 ∗-centralizeron R.

Acknowledgments The authors are greatly indebted to the referee for his\her valuable comments andsuggestions. We take this opportunity to express our gratitude to Professor Daniel Eremita for his usefuldiscussions.

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, ChicagoMartindale, 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


ORIGINAL PAPERwith involution

Documents

Transcript of ORIGINAL PAPERwith involution