• Title/Summary/Keyword: multiplicative derivation

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POSNER'S THEOREM FOR GENERALIZED DERIVATIONS ASSOCIATED WITH A MULTIPLICATIVE DERIVATION

  • UZMA NAAZ;MALIK RASHID JAMAL
    • Journal of applied mathematics & informatics
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    • v.42 no.3
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    • pp.539-548
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    • 2024
  • Let R be a ring and P be a prime ideal of R. A mapping d : R → R is called a multiplicative derivation if d(xy) = d(x)y + xd(y) for all x, y ∈ R. In this paper, our main motive is to obtain the well-known theorem due to Posner in the ring R/P for generalized derivations associated with a multiplicative derivation defined by an additive mapping F : R → R such that F(xy) = F(x)y + xd(y), where d : R → R is a multiplicative derivation not necessarily additive. This article discusses the use of generalized derivations associated with a multiplicative derivation to investigate the commutativity of the quotient ring R/P.

COMMUTATIVITY OF MULTIPLICATIVE b-GENERALIZED DERIVATIONS OF PRIME RINGS

  • Muzibur Rahman Mozumder;Wasim Ahmed;Mohd Arif Raza;Adnan Abbasi
    • Korean Journal of Mathematics
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    • v.31 no.1
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    • pp.95-107
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    • 2023
  • Consider ℛ to be an associative prime ring and 𝒦 to be a nonzero dense ideal of ℛ. A mapping (need not be additive) ℱ : ℛ → 𝒬mr associated with derivation d : ℛ → ℛ is called a multiplicative b-generalized derivation if ℱ(αδ) = ℱ(α)δ +bαd(δ) holds for all α, δ ∈ ℛ and for any fixed (0 ≠)b ∈ 𝒬s ⊆ 𝒬mr. In this manuscript, we study the commutativity of prime rings when the map b-generalized derivation satisfies the strong commutativity preserving condition and moreover, we investigate the commutativity of prime rings that admit multiplicative b-generalized derivation, which improves many results in the literature.

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.

STABILITY OF THE FUNCTIONAL EQUATIONS RELATED TO A MULTIPLICATIVE DERIVATION

  • Kim, Hark-Mahn;Chang, Ick-Soon
    • Journal of applied mathematics & informatics
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    • v.11 no.1_2
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    • pp.413-421
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    • 2003
  • In this paper, using an idea from the direct method of Hyers and Ulam, we investigate the situations so that the Hyers-Ulam-Rassias stability of the functional equation $g(x^2)\;=\;2xg(x)$ is satisfied.

Almost derivations on the banach algebra $C^n$[0,1]

  • Jun, Kil-Woung;Park, Dal-Won
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.359-366
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    • 1996
  • A linear map T from a Banach algebra A into a Banach algebra B is almost multiplicative if $\left\$\mid$ T(fg) - T(f)T(g) \right\$\mid$ \leq \in\left\$\mid$ f \right\$\mid$\left\$\mid$ g \right\$\mid$(f,g \in A)$ for some small positive $\in$. B.E.Johnson [4,5] studied whether this implies that T is near a multiplicative map in the norm of operators from A into B. K. Jarosz [2,3] raised the conjecture : If T is an almost multiplicative functional on uniform algebra A, there is a linear and multiplicative functional F on A such that $\left\$\mid$ T - F \right\$\mid$ \leq \in', where \in' \to 0$ as $\in \to 0$. B. E. Johnson [4] gave an example of non-uniform commutative Banach algebra which does not have the property described in the above conjecture. He proved also that C(K) algebras and the disc algebra A(D) have this property [5]. We extend this property to a derivation on a Banach algebra.

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On Prime Near-rings with Generalized (σ,τ)-derivations

  • Golbasi, Oznur
    • Kyungpook Mathematical Journal
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    • v.45 no.2
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    • pp.249-254
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    • 2005
  • Let N be a prime left near-ring with multiplicative center Z and f be a generalized $({\sigma},{\tau})-derivation$ associated with d. We prove commutativity theorems in prime near- rings with generalized $({\sigma},{\tau})-derivation$.

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A NOTE ON MULTIPLICATIVE (GENERALIZED)-DERIVATION IN SEMIPRIME RINGS

  • REHMAN, NADEEM UR;HONGAN, MOTOSHI
    • Journal of applied mathematics & informatics
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    • v.36 no.1_2
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    • pp.81-92
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    • 2018
  • In this article we study two Multiplicative (generalized)- derivations ${\mathcal{G}}$ and ${\mathcal{H}}$ that satisfying certain conditions in semiprime rings and tried to find out some information about the associated maps. Moreover, an example is given to demonstrate that the semiprimeness imposed on the hypothesis of the various results is essential.

Computer intensive method for extended Euclidean algorithm (확장 유클리드 알고리즘에 대한 컴퓨터 집약적 방법에 대한 연구)

  • Kim, Daehak;Oh, Kwang Sik
    • Journal of the Korean Data and Information Science Society
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    • v.25 no.6
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    • pp.1467-1474
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    • 2014
  • In this paper, we consider the two computer intensive methods for extended Euclidean algdrithm. Two methods we propose are C-programming based approach and Microsoft excel based method, respectively. Thses methods are applied to the derivation of greatest commnon devisor, multiplicative inverse for modular operation and the solution of diophantine equation. Concrete investigation for extended Euclidean algorithm with the computer intensive process is given. For the application of extended Euclidean algorithm, we consider the RSA encrytion method which is still popular recently.