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JORDAN *-DERIVATIONS OF PRIME RINGS

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Journal of Algebra and Its Applications
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Tsiu-Kwen Lee at National Taiwan University
Tsiu-Kwen Lee
Yiqiang Zhou at Memorial University of Newfoundland
Yiqiang Zhou
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Abstract

Let R be a prime ring, which is not commutative, with involution * and with Qms(R) the maximal symmetric ring of quotients of R. An additive map δ : R → R is called a Jordan *-derivation if δ(x²) = δ(x)x* + xδ(x) for all x ∈ R. A Jordan *-derivation of R is called X-inner if it is of the form x ↦ xa - ax* for x ∈ R, where a ∈ Qms(R). We prove that any Jordan *-derivation of R is X-inner if char R ≠ 2 or deg(S(R)) > 4, where S(R) := {x ∈ R|x* = x}.
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2nd Reading
November 1, 2013 10:11 WSPC/S0219-4988 171-JAA 1350126
Journal of Algebra and Its Applications
Vol. 13, No. 4 (2014) 1350126 (9pages)
c
World Scientific Publishing Company
DOI: 10.1142/S0219498813501260
JORDAN ∗-DERIVATIONS OF PRIME RINGS
TSIU-KWEN LEE∗
Department of Mathematics
National Taiwan University
Taipei 106, Taiwan
tklee@math.ntu.edu.tw
YIQIANG ZHOU
Department of Mathematics and Statistics
Memorial University of Newfound land
St. John’s, Nfld, Canada A1C 5S7
zhou@mun.ca
Received 26 June 2013
Accepted 26 August 2013
Published 5 November 2013
CommunicatedbyT.Y.Lam
Let Rbe a prime ring, which is not commutative, with involution ∗and with Qms(R)
the maximal symmetric ring of quotients of R. An additive map δ:R→Ris called a
Jordan ∗-derivation if δ(x2)=δ(x)x∗+xδ(x) for all x∈R.AJordan∗-derivation of
Ris called X-inner if it is of the form x→ xa −ax∗for x∈R,wherea∈Qms(R).
We prove that any Jordan ∗-derivation of Ris X-inner if char R=2ordeg(S(R)) >4,
where S(R):={x∈R|x∗=x}.
Keywords: Prime ring; involution; Jordan ∗-derivation; PI; maximal symmetric ring of
quotients; functional identity.
Mathematics Subject Classification: 16R60, 16N60, 16W10, 16R50
1. Results
Throughout the paper, Ralways denotes an associative prime ring with center
Z(R). We write Qml(R) for the maximal left ring of quotients of Rand Qms(R)
for the maximal symmetric ring of quotients of R.ItisknownthatR⊆Qms(R)⊆
Qml(R). The two overrings of Rare also prime rings with the same center C,which
is a field. The field Cis called the extended centroid of R. We refer the reader to
the book [3] for details.
∗Member of Mathematics Division, NCTS (Taipei Office).
1350126-1
... Šemrl studied the Jordan * -derivations and quadratic forms making use of bilinear forms, see [21] and [20]. Lee et al. in [13], Theorem 2.12, [15] and [16] obtained that any Jordan * -derivation on a noncommutative prime ring A with an involution ' * ' is X-inner unless dim C AC = 4 and char(A) = 2. Vukman in [27], Theorem 4 first characterized (m, n)-Jordan centralizers in prime rings and proved that every (m, n)-Jordan centralizer on a prime ring A is a two sided centralizer provided Z(A) is nonzero and A is 6mn(m + n)-torsion free. Fošner [7], proved that any generalized (m, n)-Jordan centralizer on a 6mn(m+n)(m+2n)-torsion free prime ring A is a two sided centralizer provided that Z(A) is nonzero. ...
... We begin with the following results which play a pivotal role in the proof of our main results. The first result holds for any arbitrary ring A, see [16], Lemma 2.3. P r o o f . ...
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... Semrl studied the Jordan * -derivation and quadratic forms making use of bilinear forms (see [20,21]). Lee et al. [14,15] obtained that any Jordan * -derivation on a noncommutative prime ring A with involution ' * ' is X-inner unless dim C AC = 4 and char(A) = 2. Lee and Wong [13] proved that any Jordan τ-derivation of a noncommutative prime ring A is X-inner if either A is not a GPI-ring or A is a PI-ring except when char(A) = 2 and dim C AC = 4. Very recently Lin [16] showed that when A is a prime GPI-ring but not a PI-ring, then any Jordan τ-derivation of A is X-inner if either τ is of the second kind or both char(A) = 2 and τ is of the first kind with deg τ 2 = 2. Vukman [22] proved that if m and n are distinct positive integers such that A is a 2mn(m + n)|m − n|-torsion free prime ring, then any nonzero (m, n)-Jordan derivation F : A → A is a derivation and A is commutative unless char(A) = 2 or 3. Ali and Fošner [1] generalized this result and proved that if A is a 6mn(m + n)|m − n|-torsion free prime ring and G : A → A is a nonzero generalized (m, n)-Jordan derivation, then G is a derivation and A is commutative provided that char(A) = 0 or char(A) > 3. Recently, Bennis, Dhara and Fahid [4, Theorem 1.3] characterized nonzero generalized (m, n)-Jordan derivations in semiprime rings and obtained that if A is a 6mn(m + n)|m − n|-torsion free semiprime ring, then any nonzero generalized (m, n)-Jordan derivation on A is a derivation and maps A into Z(A). For other results see [6][7][8]17]. ...
... By [15,Lemma 2.3], it follows that δ satisfies the following relation: ...
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... For instance, Haetinger and Mamouni [1] introduced the concept of generalized left semi-derivations in the non-commutative ring and presented some results about it. A paper by Lee and Zhou [2] had studied the notion of Jordan -derivations of prime rings. The authors showed that any Jordan -derivation of non-commutative prime ring with involution is X-inner. ...
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