• Title/Summary/Keyword: (generalized) derivation

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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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b-GENERALIZED DERIVATIONS ON MULTILINEAR POLYNOMIALS IN PRIME RINGS

  • Dhara, Basudeb
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.2
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    • pp.573-586
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    • 2018
  • Let R be a noncommutative prime ring of characteristic different from 2, Q be its maximal right ring of quotients and C be its extended centroid. Suppose that $f(x_1,{\ldots},x_n)$ be a noncentral multilinear polynomial over $C,b{\in}Q,F$ a b-generalized derivation of R and d is a nonzero derivation of R such that d([F(f(r)), f(r)]) = 0 for all $r=(r_1,{\ldots},r_n){\in}R^n$. Then one of the following holds: (1) there exists ${\lambda}{\in}C$ such that $F(x)={\lambda}x$ for all $x{\in}R$; (2) there exist ${\lambda}{\in}C$ and $p{\in}Q$ such that $F(x)={\lambda}x+px+xp$ for all $x{\in}R$ with $f(x_1,{\ldots},x_n)^2$ is central valued in R.

LEFT JORDAN DERIVATIONS ON BANACH ALGEBRAS AND RELATED MAPPINGS

  • Jung, Yong-Soo;Park, Kyoo-Hong
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.1
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    • pp.151-157
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    • 2010
  • In this note, we obtain range inclusion results for left Jordan derivations on Banach algebras: (i) Let $\delta$ be a spectrally bounded left Jordan derivation on a Banach algebra A. Then $\delta$ maps A into its Jacobson radical. (ii) Let $\delta$ be a left Jordan derivation on a unital Banach algebra A with the condition sup{r$(c^{-1}\delta(c))$ : c $\in$ A invertible} < $\infty$. Then $\delta$ maps A into its Jacobson radical. Moreover, we give an exact answer to the conjecture raised by Ashraf and Ali in [2, p. 260]: every generalized left Jordan derivation on 2-torsion free semiprime rings is a generalized left derivation.

ON GENERALIZED DERIVATIONS OF BE-ALGEBRAS

  • Kim, Kyung Ho
    • Journal of the Chungcheong Mathematical Society
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    • v.27 no.2
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    • pp.227-236
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    • 2014
  • In this paper, we introduce the notion of a generalized derivation in a BE-algebra, and consider the properties of generalized derivations. Also, we characterize the fixed set $Fix_d(X)$ and Kerd by generalized derivations. Moreover, we prove that if d is a generalized derivation of a BE-algebra, every filter F is a d-invariant.

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.