• Journal of the Chungcheong Mathematical Society
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• v.23 no.1
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• pp.49-57
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• 2010

It is shown that for a derivation $$f(x_1{\cdots}x_{j-1}x_jx_{j+1}{\cdots}x_k)=\sum_{j=1}^{k}x_{1}{\cdots}x_{j-1}x_{j+1}{\cdots}x_kf(x_j)$$ on a unital $C^*$-algebra $\mathcal{B}$, there exists a unique $\mathbb{C}$-linear *-derivation $D:{\mathcal{B}}{\rightarrow}{\mathcal{B}}$ near the derivation, by using the Hyers-Ulam-Rassias stability of functional equations. The concept of Hyers-Ulam-Rassias stability originated from the Th.M. Rassias' stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1123-1132
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• 2015

In this paper, we investigate derivations on the noncommutative Banach algebra $L^{\infty}_0({\omega})^*$ equipped with an Arens product. As a main result, we prove the Singer-Wermer conjecture for the noncommutative Banach algebra $L^{\infty}_0({\omega})^*$. We then show that a derivation on $L^{\infty}_0({\omega})^*$ is continuous if and only if its restriction to rad($L^{\infty}_0({\omega})^*$) is continuous. We also prove that there is no nonzero centralizing derivation on $L^{\infty}_0({\omega})^*$. Finally, we prove that the space of all inner derivations of $L^{\infty}_0({\omega})^*$ is continuously homomorphic to the space $L^{\infty}_0({\omega})^*/L^1({\omega})$.

• Communications of the Korean Mathematical Society
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• v.31 no.2
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• pp.247-259
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• 2016

We first give the constructions of (Jordan) higher derivations on a trivial extension algebra and then we provide some sufficient conditions under which a Jordan higher derivation on a trivial extension algebra is a higher derivation. We then proceed to the trivial generalized matrix algebras as a special trivial extension algebra. As an application we characterize the construction of Jordan higher derivations on a triangular algebra. We also provide some illuminating examples of Jordan higher derivations on certain trivial extension algebras which are not higher derivations.

• Korean Journal of Mathematics
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• v.24 no.3
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• pp.345-367
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• 2016

The concept of derivations of BCI-algebras was first introduced by Jun and Xin. In this paper, we introduce the notions of (l, r)-derivations, (r, l)-derivations and derivations of UP-algebras and investigate some related properties. In addition, we define two subsets $Ker_d(A)$ and $Fix_d(A)$ for some derivation d of a UP-algebra A, and we consider some properties of these as well.

• East Asian mathematical journal
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• v.21 no.1
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• pp.1-7
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• 2005

The object of this paper is to study($\sigma,\;\tau$)-Lie ideals in semi-prime rings with derivation. Main result is the following theorem: Let R be a semi-prime ring with 2-torsion free, $\sigma$ and $\tau$ two automorphisms of R such that $\sigma\tau=\tau\sigma$=, U be both a non-zero ($\sigma,\;\tau$)-Lie ideal and subring of R. If $d^2(U)=0$, then d(U)=0 where d a non-zero derivation of R such that $d\sigma={\sigma}d,\;d\tau={\tau}d$.

• East Asian mathematical journal
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• v.18 no.2
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• pp.195-203
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• 2002

Let R be a prime ring with characteristic not two and U is a nonzero left ideal of R which contains no nonzero nilpotent right ideal as a ring. For a $(\sigma,\tau)$-derivation d : R$\rightarrow$R, we prove the following results: (1) If d is an endomorphism on R then d=0. (2) If d is an anti-endomorphism on R then d=0. (3) If d(xy)=d(yx), for all x, y$\in$R then R is commutative. (4) If d is an homomorphism or anti-homomorphism on U then d=0.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.1
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• pp.1-12
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• 2018

Let R be a 3!-torsion free noncommutative semiprime ring, and suppose there exists a Jordan derivation $D:R{\rightarrow}R$ such that $D^2(x)[D(x),x]=0$ or $[D(x),x]D^2(x)=0$ for all $x{\in}R$. In this case we have $f(x)^5=0$ for all $x{\in}R$. Let A be a noncommutative Banach algebra. Suppose there exists a continuous linear Jordan derivation $D:A{\rightarrow}A$ such that $D^2(x)[D(x),x]{\in}rad(A)$ or $[D(x),x]D^2(x){\in}rad(A)$ for all $x{\in}A$. In this case, we show that $D(A){\subseteq}rad(A)$.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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• pp.103-125
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• 2018

Let R be a 5!-torsion free semiprime ring, and let $D:R{\rightarrow}R$ be a Jordan derivation on a semiprime ring R. Then $[D(x),x]D(x)^2=0$ if and only if $D(x)^2[D(x), x]=0$ for every $x{\in}R$. In particular, let A be a Banach algebra with rad(A) and if D is a continuous linear Jordan derivation on A, then we show that $[D(x),x]D(x)2{\in}rad(A)$ if and only if $D(x)^2[D(x),x]{\in}rad(A)$ for all $x{\in}A$ where rad(A) is the Jacobson radical of A.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1209-1219
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• 2021

Let G be a topological group with a locally compact and Hausdorff topology. Let ω be a diagonally bounded weight on G. In this paper, (σ, σ)-derivation and (σ, 𝜏)-weak amenability of the Beurling algebra L¹_ω(G) are studied, where σ, 𝜏 are isometric automorphisms of L¹_ω(G). We prove that every continuous (σ, σ)-derivation from L¹_ω(G) into measure algebra M_ω(G) is (σ, σ)-inner and the Beurling algebra L¹_ω(G) is (σ, 𝜏)-weakly amenable.

• Communications of the Korean Mathematical Society
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• v.36 no.2
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• pp.229-238
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• 2021

Let R be a ring with characteristic different from 2. An additive mapping F : R → R is called a generalized derivation on R if there exists a derivation d : R → R such that F(xy) = F(x)y + xd(y) holds for all x, y ∈ R. In the present paper, we show that if R is a prime ring satisfying certain identities involving a generalized derivation F associated with a derivation d, then R becomes commutative and in some cases d comes out to be zero (i.e., F becomes a left centralizer). We provide some counter examples to justify that the restrictions imposed in the hypotheses of our theorems are not superfluous.