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