• Bulletin of the Korean Mathematical Society
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• v.43 no.3
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• pp.627-634
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• 2006

A Weyl type algebra is defined in the book ([4]). A Weyl type non-associative algebra $\={WP_{m,n,s}}$ and its restricted sub-algebra $\={WP_{m,n,s_{\gamma}}}$ are defined in various papers ([1], [12], [3], [11]). Several authors 0nd all the derivations of an associative (Lie or non-associative) algebra in the papers ([1], [2], [12], [4], [6], [11]). We find all the non-associative algebra derivations of the non-associative algebra $\={WP_{0,2,0_B}$, where $B=\{{\partial}_0,\;{\partial}_1,\;{\partial}_2,\;{\partial}_{12},\;{\partial}^2_1,\;{\partial}^2_2\}$.

• Kyungpook Mathematical Journal
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• v.56 no.2
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• pp.397-408
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• 2016

Let R be a 2-torsion free prime ring with center Z, U be the Utumi quotient ring, Q be the Martindale quotient ring of R, d be a derivation of R and L be a Lie ideal of R. If $d(uv)^n=d(u)^md(v)^l$ or $d(uv)^n=d(v)^ld(u)^m$ for all $u,v{\in}L$, where m, n, l are xed positive integers, then $L{\subseteq}Z$. We also examine the case when R is a semiprime ring. Finally, as an application we apply our result to the continuous derivations on non-commutative Banach algebras. This result simultaneously generalizes a number of results in the literature.

• Honam Mathematical Journal
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• v.30 no.2
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• pp.273-281
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• 2008

We define the Lie-admissible algebra NW$({\mathbb{F}}[e^{A[s]},x_1,{\cdots},x_n])$ in this work. We show that the algebra and its antisymmetrized (i.e., Lie) algebra are simple. We also find all the derivations of the algebra NW$(F[e^{{\pm}x^r},x])$ and its antisymmetrized algebra W$(F[e^{{\pm}x^r},x])$ in the paper.

• Honam Mathematical Journal
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• v.36 no.1
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• pp.179-186
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• 2014

In this paper, we consider the simple non-associative algebra $\overline{WN(\mathbb{F}[e^{{\pm}x^r},0,1]_{(\partial,\partial^2)})}$. There are many papers on finding the derivations of an associative algebra, a Lie algebra, and a non-associative algebra (see [2], [3], [4], [5], [6], [7], [12], [14]). We find all the derivations of the algebra $\overline{WN(\mathbb{F}[e^{{\pm}x^r},0,1]_{(\partial,\partial^2)})}$.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.73-83
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• 2018

Let R be an associative ring with involution * and ${\alpha},{\beta}:R{\rightarrow}R$ ring homomorphisms. An additive mapping $d:R{\rightarrow}R$ is called an $({\alpha},{\beta})^*$-derivation of R if $d(xy)=d(x){\alpha}(y^*)+{\beta}(x)d(y)$ is fulfilled for any $x,y{\in}R$, and an additive mapping $F:R{\rightarrow}R$ is called a generalized $({\alpha},{\beta})^*$-derivation of R associated with an $({\alpha},{\beta})^*$-derivation d if $F(xy)=F(x){\alpha}(y^*)+{\beta}(x)d(y)$ is fulfilled for all $x,y{\in}R$. In this note, we intend to generalize a theorem of Vukman [12], and a theorem of Daif and El-Sayiad [6], moreover, we generalize a theorem of Ali et al. [4] and a theorem of Huang and Koc [9] related to generalized Jordan triple $({\alpha},{\beta})^*$-derivations.

• Korean Journal of Mathematics
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• v.24 no.4
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• pp.587-600
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• 2016

We obtain the general solution of the functional equation $f(ax+y)-f(x-ay)+{\frac{1}{2}}a(a^2+1)f(x-y)+(a^4-1)f(y)={\frac{1}{2}}a(a^2+1)f(x+y)+(a^4-1)f(x)$ and prove the stability problem of the quartic Lie ${\ast}$-derivation by using a directed method and an alternative fixed point method.

• Journal for History of Mathematics
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• v.10 no.2
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• pp.24-29
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• 1997

Derivation은 Lie group, Lie ring 그리고 Lie Algebra에서 정의되어 사용되며 발전하였으며 ring에서 일반화 되었다. 역시 prime ring에서 연구되어지는 derivation의 성질들은 prime near-ring에서 일반화 시키려고 하고 있다. 1957년 E. Posner는 prime ring에서 두 개의 derivation의 곱(함수합성)이 derivation이면 이들중 하나의 derivation이 0임을 밝혔다. 본 논문에서는 prime ring에서 derivation이 연구된 역사적인 배경을 소개하고 몇가지 성질을 찾는다. 즉, D. F를 prime ring R의 derivation들이라 할 때 정수 $n{\ge}1$에 대하여 $DF^n$=0이면 D=0이거나 또는 $F^{3n-1}$=0이고, $D^nF$=0이면 $D^{9n-7}$=0 이거나 또는 $F^2$=0 이다.

• Communications of the Korean Mathematical Society
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• v.32 no.3
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• pp.535-542
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• 2017

Let R be an associative ring and ${\alpha},{\beta}:R{\rightarrow}R$ ring homomorphisms. An additive mapping $d:R{\rightarrow}R$ is called an (${\alpha},{\beta}$)-derivation of R if $d(xy)=d(x){\alpha}(y)+{\beta}(x)d(y)$ is fulfilled for any $x,y{\in}R$, and an additive mapping $D:R{\rightarrow}R$ is called a generalized (${\alpha},{\beta}$)-derivation of R associated with an (${\alpha},{\beta}$)-derivation d if $D(xy)=D(x){\alpha}(y)+{\beta}(x)d(y)$ is fulfilled for all $x,y{\in}R$. In this note, we intend to generalize a theorem of Vukman [5], and a theorem of Daif and El-Sayiad [2].

• Korean Journal of Mathematics
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• v.27 no.4
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• pp.969-976
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• 2019

Let ℬ(ℋ) be the algebra of all bounded linear operators on a complex Hilbert space ℋ and 𝒨 ⊆ ℬ(ℋ) be a von Neumann algebra without central abelian projections. Let ξ be a non-zero scalar. In this paper, it is proved that a mapping φ : 𝒨 → ℬ(ℋ) satisfies φ([A, B]^ξ_⁎)= [φ(A), B]^ξ_⁎+[A, φ(B)]^ξ_⁎ for all A, B ∈ 𝒨 if and only if φ is an additive ⁎-derivation and φ(ξA) = ξφ(A) for all A ∈ 𝒨.

• 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.