Generalized Derivations on (Semi-)Prime Rings and Noncommutative Banach Algebras

Wei, Feng; Xiao, Zhongmin

doi:10.4171/rsmup/122-11

Cited by 5 publications

(2 citation statements)

References 26 publications

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“…Then in [27] Vukman proved that if m, n ≥ 1 are fixed integers and there exist continuous derivations D, G of a Banach algebra R such that D(x m )x n − x n G(x m ) ∈ rad(R) for all x ∈ R, then both D and G map into rad(R). A similar result was obtained by Wei and Xiao in [28] in case D and G are continuous Jordan derivations.…”

Section: Introductionsupporting

confidence: 86%

Identities with Generalized Derivations on Prime Rings and Banach Algebras

Carini

Filippis

2012

Algebra Colloq.

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Let R be a prime ring, U the Utumi quotient ring of R, C = Z(U ) the extended centroid of R, L a non-central Lie ideal of R, and H, G nonzero generalized derivations of R. Suppose that there exists an integer n ≥ 1 such that H(u n )u n + u n G(u n ) ∈ C for all u ∈ L, then either there exists a ∈ U such that H(x) = xa and G(x) = −ax, or R satisfies the standard identity s4 and one of the following holds: (i) char(R) = 2; (ii) n is even and there exist a ∈ U , α ∈ C and derivations d, δ of R such that H(x) = a x + d(x) and G(x) = (α − a )x + δ(x); (iii) n is even and there exist a ∈ U and a derivation δ of R such that H(x) = xa and G(x) = −a x + δ(x); (iv) n is odd and there exist a , b ∈ U and α, β ∈ C such that H(x) = a x + x(β − b ) and G(x) = b x + x(α − a ); (v) n is odd and there exist α, β ∈ C and a derivation d of R such that H(x) = αx + d(x) and G(x) = βx + d(x); (vi) n is odd and there exist a ∈ U and α ∈ C such that H(x) = xa and G(x) = (α − a )x. As an application of this purely algebraic result, we obtain some range inclusion results of continuous or spectrally bounded generalized derivations H and

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Section: Introductionsupporting

confidence: 86%

Identities with Generalized Derivations on Prime Rings and Banach Algebras

Carini

Filippis

2012

Algebra Colloq.

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“…There has been considerable work which were contributed to the study of generalized Jordan derivations on prime rings, see [1], [2], [4], [9], [15], [17], [20], [28], [30]. One of current research topics is to consider when a generalized Jordan derivation on a prime ring can degenerate a zero mapping or a left(or right) multiplier.…”

Section: Degenerating Generalized Jordan Derivations On Prime Ringsmentioning

confidence: 99%

Generalized Jordan Derivations on Semiprime Rings and Its Applications in Range Inclusion Problems

Wei

Xiao

2010

Mediterr. J. Math.

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It is shown that any generalized Jordan (triple-)derivation on a 2-torsion free semiprime ring is a generalized derivation and that any generalized Jordan higher derivation on a 2-torsion free semiprime ring is a generalized higher derivation. Then we give several conditions which enable some generalized Jordan derivations on prime rings to degenerate left or right multipliers. Lastly, we apply these degenerating conditions to discuss the range inclusion problems of generalized derivations on noncommutative Banach algebras.Mathematics Subject Classification (2010). Primary 16W25, 16N60; Secondary 47B47.

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