On (m,n)-Jordan centralizers in rings and algebras

Vukman, Joso

doi:10.3336/gm.45.1.04

Cited by 15 publications

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“…In [8], Vukman proved that on a prime ring with a nonzero center Z(R) and char(R) = 6mn(m + n) every (m, n)-Jordan centralizer is a two-sided centralizer. The natural question here is whether an analogue holds true for generalized (m, n)-Jordan centralizers.…”

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A note on generalized (m, n)-Jordan centralizers

Fošner

2013

Demonstratio Mathematica

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Abstract. The aim of this paper is to define generalized (m, n)-Jordan centralizers and to prove that on a prime ring with nonzero center and char(R) = 6mn(m+n)(m+2n) every generalized (m, n)-Jordan centralizer is a two-sided centralizer.Throughout, R will represent an associative ring with a center Z(R). Let n ≥ 2 be an integer. A ring R is said to be n-torsion free if for x ∈ R, nx = 0 implies x = 0. Recall that R is prime if aRb = {0} implies a = 0 oris fulfilled for all x ∈ R. One can easily prove that every derivation is a Jordan derivation, but converse is in general not true. A classical result due to Herstein [5, Theorem 3.3] asserts that a Jordan derivation on a prime ring of characteristic different from two is a derivation. A brief proof of Herstein's result can be found in [3]. This result was extended to 2-torsion free semiprime rings by Cusack [4] (see [2] for an alternative proof).An additive mapping T : R → R is called a left (right) centralizer if T (xy) = T (x)y (T (xy) = xT (y)) holds for all pairs x, y ∈ R. If R has the 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, where a ∈ R is a fixed element. For a semiprime ring R, left centralizers are of the form T (x) = qx for all x ∈ R, where q is a fixed element of a Martindale right ring of quotients Q r (see, for example, Chapter 2 in [1]). An additive mapping T : R → R is called a left (right) Jordan centralizer if T (x 2 ) = T (x)x (T (x 2 ) = xT (x)) holds for all x ∈ R. We call an additive mapping T : R → R a two-sided centralizer 2010 Mathematics Subject Classification: 16N60, 39B05.

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“…Let F : R → R be an additive mapping defined by F (x) = T (x) − T 0 (x), x ∈ R. We would like to show that F (x) = 0 for all x ∈ R. Namely, if F (x) = T (x) − T 0 (x) = 0, then T (x) = T 0 (x) for all x ∈ R, which yields that T is a two-sided centralizer, since T 0 is a two-sided centralizer by [8,Theorem 2].…”

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A note on generalized (m, n)-Jordan centralizers

Fošner

2013

Demonstratio Mathematica

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“…Zalar ([20]) has proved that any left (right) Jordan centralizer on a semiprime ring is a left (right) centralizer. For results concerning centralizers in rings and algebras we refer to [11,15,[17][18][19][20] where further references can be found. A mapping F, which maps a ring R into itself, is called centralizing on R in case [F (x), x] ∈ Z(R) holds for all x ∈ R. A classical result of Posner ( [13]) (Posner's second theorem) states that the existence of a nonzero centralizing derivation on a prime ring R with char(R) = 2 forces the ring to be commutative.…”

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“…for all x ∈ R (see [19] for the details). Recently, Vukman ([19]) considered the above relation in standard operator algebras on a real or complex Hilbert space.…”

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On certain functional equation arising from (m,n)- Jordan centralizers in prime rings

Peršin¹,

Vukman²

2012

Glas. Mat. Ser. III

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Abstract. The purpose of this paper is to prove the following result. Let m ≥ 1, n ≥ 1 be some fixed integers and let R be a prime ring with char(R) = 0 or (m + n) 2 < char(R). Suppose there exists an additive mapping T : R → R satisfying the relation 2(m + n) 2 T (x 3 ) = m(2m + n)T (x)x 2 + 2mnxT (x)x + n(2n + m)x 2 T (x) for all x ∈ R. In this case T is a two-sided centralizer.Throughout, R will represent an associative ring with center Z(R). 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 the commutator xy − yx will be denoted by [x, y]. We shall use the commutator identities [xy, z] Recall that a ring R is prime if for a, b ∈ R, aRb = (0) implies that either a = 0 or b = 0 and is semiprime in case aRa = (0) implies a = 0. We denote by char(R) the characteristic of a prime ring R. An additive mapping D : R → R, where R is an arbitrary ring, is called a derivation if D(xy) = D(x)y + xD(y) holds for all pairs x, y ∈ R, and is called a Jordan derivation in case D(x 2 ) = D(x)x + xD(x) is fulfilled for all x ∈ R. A derivation D is inner in case there exists a ∈ R, such that D(x) = [a, x] holds for all x ∈ R. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein ([10]) asserts that any Jordan derivation on a prime ring with char(R) = 2 is a derivation. A brief proof of Herstein's result can be found in [3]. Cusack 2010 Mathematics Subject Classification. 16W10, 46K15, 39B05.

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“…φ(A 2 ) = Aφ(A)) for any A ∈ R. A Jordan centralizer of R is an additive mapping which is a left Jordan as well as a right Jordan centralizer. An (m, n)− Jordan centralizer is defined in ( [16]) as follows: An additive mapping φ : R → R is called an (m, n)− Jordan centralizer if (m+ n)φ(A 2 ) = mφ(A)A+ nAφ(A) for any A ∈ R, where m, n ∈ N with m+ n = 0. Obviously, every centralizer is a Jordan centralizer and any Jordan centralizer is an (m, n)− Jordan centralizer, but the converse is not true in general.…”

Section: Introductionmentioning

confidence: 99%

The characterization of generalized Jordan centralizers on algebras

Chen

Fang

2017

B Soc Paran Mat

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In this paper, it is shown that if A is a CSL subalgebra of a von Neumann algebr and φ is a continuous mapping on A such that (m + n + k + l)φ(A 2 ) − (mφ(A)A + nAφ(A) + kφ(I)A 2 + lA 2 φ(I)) ∈ FI for any A ∈ A, where F is the real field or the complex field, then φ is a centralizer. It is also shown that if φ is an additive mapping on A such that (m + n + k + l)φ(A 2 ) = mφ(A)A + nAφ(A) + kφ(I)A 2 + lA 2 φ(I) for any A ∈ A, then φ is a centralizer.

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On (m,n)-Jordan centralizers in rings and algebras

Abstract: Abstract. Let m ≥ 0, n ≥ 0 be fixed integers with m + n = 0 and let R be a ring. It is our aim in this paper to investigate additive mapping T : R → R satisfying the relation (m + n)T (x 2 ) = mT (x)x + nxT (x) for all x ∈ R.

Cited by 15 publications

References 14 publications

A note on generalized (m, n)-Jordan centralizers

A note on generalized (m, n)-Jordan centralizers

On certain functional equation arising from (m,n)- Jordan centralizers in prime rings

The characterization of generalized Jordan centralizers on algebras

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