• Title/Summary/Keyword: (generalized) derivation

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ON GENERALIZED (σ, τ)-DERIVATIONS II

  • Argac, Nurcan;Inceboz, Hulya G.
    • Journal of the Korean Mathematical Society
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    • v.47 no.3
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    • pp.495-504
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    • 2010
  • This paper continues a line investigation in [1]. Let A be a K-algebra and M an A/K-bimodule. In [5] Hamaguchi gave a necessary and sufficient condition for gDer(A, M) to be isomorphic to BDer(A, M). The main aim of this paper is to establish similar relationships for generalized ($\sigma$, $\tau$)-derivations.

Correction to "On prime near-rings with generalized (σ, τ)- derivations, Kyungpook Math. J., 45(2005), 249-254"

  • Al Hwaeer, Hassan J.;Albkwre, Gbrel;Turgay, Neset Deniz
    • Kyungpook Mathematical Journal
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    • v.60 no.2
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    • pp.415-421
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    • 2020
  • In the proof of Theorem 3 on p.253 in [4], both right and left distributivity are assumed simultaneously which makes the proof invalid. We give a corrected proof for this theorem by introducing an extension of Lemma 2.2 in [2].

MULTIPLICATIVE (GENERALIZED) (𝛼, 𝛽)-DERIVATIONS ON LEFT IDEALS IN PRIME RINGS

  • SHUJAT, FAIZA
    • Journal of Applied and Pure Mathematics
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    • v.4 no.1_2
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    • pp.1-7
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    • 2022
  • A mapping T : R → R (not necessarily additive) is called multiplicative left 𝛼-centralizer if T(xy) = T(x)𝛼(y) for all x, y ∈ R. A mapping F : R → R (not necessarily additive) is called multiplicative (generalized)(𝛼, 𝛽)-derivation if there exists a map (neither necessarily additive nor derivation) f : R → R such that F(xy) = F(x)𝛼(y) + 𝛽(x)f(y) for all x, y ∈ R, where 𝛼 and 𝛽 are automorphisms on R. The main purpose of this paper is to study some algebraic identities with multiplicative (generalized) (𝛼, 𝛽)-derivations and multiplicative left 𝛼-centralizer on the left ideal of a prime ring R.

GENERALIZED (𝜃, 𝜙)-DERIVATIONS ON POISSON BANACH ALGEBRAS AND JORDAN BANACH ALGEBRAS

  • Park, Chun-Gil
    • Journal of the Chungcheong Mathematical Society
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    • v.18 no.2
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    • pp.175-193
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    • 2005
  • In [1], the concept of generalized (${\theta}$, ${\phi}$)-derivations on rings was introduced. In this paper, we introduce the concept of generalized (${\theta}$, ${\phi}$)-derivations on Poisson Banach algebras and of generalizd (${\theta}$, ${\phi}$)-derivations on Jordan Banach algebras, and prove the Cauchy-Rassias stability of generalized (${\theta}$, ${\phi}$)-derivations on Poisson Banach algebras and of generalized (${\theta}$, ${\phi}$)-derivations on Jordan Banach algebras.

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On Semiprime Rings with Generalized Derivations

  • Khan, Mohd Rais;Hasnain, Mohammad Mueenul
    • Kyungpook Mathematical Journal
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    • v.53 no.4
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    • pp.565-571
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    • 2013
  • In this paper, we investigate the commutativity of a semiprime ring R admitting a generalized derivation F with associated derivation D satisfying any one of the properties: (i) $F(x){\circ}D(y)=[x,y]$, (ii) $D(x){\circ}F(y)=F[x,y]$, (iii) $D(x){\circ}F(y)=xy$, (iv) $F(x{\circ}y)=[F(x) y]+[D(y),x]$, and (v) $F[x,y]=F(x){\circ}y-D(y){\circ}x$ for all x, y in some appropriate subsets of R.

An Asymmetric Fuglede-Putnam's Theorem and Orthogonality

  • Ahmed, Bachir;Segres, Abdelkder
    • Kyungpook Mathematical Journal
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    • v.46 no.4
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    • pp.497-502
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    • 2006
  • An asymmetric Fuglede-Putnam theorem for $p$-hyponormal operators and class ($\mathcal{Y}$) is proved, as a consequence of this result, we obtain that the range of the generalized derivation induced by the above classes of operators is orthogonal to its kernel.

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GENERALIZED DERIVATIONS ON SEMIPRIME RINGS

  • De Filippis, Vincenzo;Huang, Shuliang
    • Bulletin of the Korean Mathematical Society
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    • v.48 no.6
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    • pp.1253-1259
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    • 2011
  • Let R be a prime ring, I a nonzero ideal of R and n a fixed positive integer. If R admits a generalized derivation F associated with a derivation d such that c for all x, $y{\in}I$. Then either R is commutative or n = 1, d = 0 and F is the identity map on R. Moreover in case R is a semiprime ring and $(F([x,\;y]))^n=[x,\;y]$ for all x, $y{\in}R$, then either R is commutative or n = 1, $d(R){\subseteq}Z(R)$, R contains a non-zero central ideal and for all $x{\in}R$.