• 대한수학회보
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• 제46권3호
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• pp.435-437
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• 2009

In this paper we prove that every Jordan $\theta$-derivation on a Lie triple system is a $\theta$-derivation. Specially, we conclude that every Jordan derivation on a Lie triple system is a derivation.

• 대한수학회지
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• 제49권2호
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• pp.419-433
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• 2012

In this paper, we show that under certain conditions every Lie higher derivation and Lie triple derivation on a triangular algebra are proper, respectively. The main results are then applied to (block) upper triangular matrix algebras and nest algebras.

• 대한수학회보
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• 제52권2호
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• pp.581-591
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• 2015

Let $\mathcal{A}$ be a factor von Neumann algebra with dimension greater than 1. We prove that if a linear map ${\delta}:\mathcal{A}{\rightarrow}\mathcal{A}$ satisfies $${\delta}([[a,b],c])=[[{\delta}(a),b],c]+[[a,{\delta}(b),c]+[[a,b],{\delta}(c)]$$ for any $a,b,c{\in}\mathcal{A}$ with ab = 0 (resp. ab = P, where P is a fixed nontrivial projection of $\mathcal{A}$), then there exist an operator $T{\in}\mathcal{A}$ and a linear map $f:\mathcal{A}{\rightarrow}\mathbb{C}I$ vanishing at every second commutator [[a, b], c] with ab = 0 (resp. ab = P) such that ${\delta}(a)=aT-Ta+f(a)$ for any $a{\in}\mathcal{A}$.

• 대한수학회논문집
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• 제33권1호
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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.

• Kyungpook Mathematical Journal
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• 제60권4호
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• pp.683-710
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• 2020

Let ℕ be the set of nonnegative integers and 𝕬 be a 2-torsion free triangular algebra over a commutative ring ℛ. In the present paper, under some lenient assumptions on 𝕬, it is proved that if Δ = {𝛿_n}_n∈ℕ is a sequence of ℛ-linear mappings 𝛿_n : 𝕬 → 𝕬 satisfying ${\delta}_n([[x,\;y],\;z])\;=\;\displaystyle\sum_{i+j+k=n}\;[[{\delta}_i(x),\;{\delta}_j(y)],\;{\delta}_k(z)]$ for all x, y, z ∈ 𝕬 with xy = 0 (resp. xy = p, where p is a nontrivial idempotent of 𝕬), then for each n ∈ ℕ, 𝛿_n = d_n + 𝜏_n; where d_n : 𝕬 → 𝕬 is ℛ-linear mapping satisfying $d_n(xy)\;=\;\displaystyle\sum_{i+j=n}\;d_i(x)d_j(y)$ for all x, y ∈ 𝕬, i.e. 𝒟 = {d_n}_n∈ℕ is a higher derivation on 𝕬 and 𝜏_n : 𝕬 → Z(𝕬) (where Z(𝕬) is the center of 𝕬) is an ℛ-linear map vanishing at every second commutator [[x, y], z] with xy = 0 (resp. xy = p).