• 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.36 no.2
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• pp.247-257
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• 1999

In this article, we give an example of local derivation, that is not derivation, on the algebra F(x1,…, xn) of rational functions in x1, …, xn over an infinite field F, and show that if X is a set of symbols and {x1,…, xn} is a finite subset of X, n$\geq$1, then each local derivation of F[x1,…, xn] into F[X] is a F-derivation and each local derivation of F[X] into itself is also a F-derivation.

• The Pure and Applied Mathematics
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• v.23 no.4
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• pp.347-375
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• 2016

Let R be a 3!-torsion free noncommutative semiprime ring, U a Lie ideal of R, and let $D:R{\rightarrow}R$ be a Jordan derivation. If [D(x), x]D(x) = 0 for all $x{\in}U$, then D(x)[D(x), x]y - yD(x)[D(x), x] = 0 for all $x,y{\in}U$. And also, if D(x)[D(x), x] = 0 for all $x{\in}U$, then [D(x), x]D(x)y - y[D(x), x]D(x) = 0 for all $x,y{\in}U$. And we shall give their applications in Banach algebras.

• Communications of the Korean Mathematical Society
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• v.22 no.4
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• pp.487-491
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• 2007

In this paper, we introduce a symmetric $bi-({\sigma},\;{\tau})-derivation$ in a near-ring and generalize some of the results in [5, 6, 8, 9].

• Bulletin of the Korean Mathematical Society
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• v.43 no.1
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• pp.101-106
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• 2006

Let R be a prime ring and I a nonzero ideal of R. Let $\alpha,\;\nu,\;\tau\;R{\rightarrow}R$ be the endomorphisms and $\beta,\;\mu\;R{\rightarrow}R$ the automorphisms. If R admits a generalized $(\alpha,\;\beta)-derivation$ g associated with a nonzero $(\alpha,\;\beta)-derivation\;\delta$ such that $g([\mu(x),y])\;=\;[\nu/(x),y]\alpha,\;\tau$ for all x, y ${\in}I$, then R is commutative.

• Honam Mathematical Journal
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• v.34 no.1
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• pp.55-62
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• 2012

In this article, we take account of stability for ring Jordan left derivations and ring left derivations and we also deal with problems for the radical ranges of linear Jordan left derivations and linear left derivations.

• Journal of the Korean Ceramic Society
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• v.14 no.4
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• pp.242-247
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• 1977

By means of coloring of plane nets a number of different tetrahedral frameworks can be derived. With the aid of this method all tetrahedral framework structures which built up of layer structures with 2, 4, 8 tetrahedra per unit cell and 6-membered ring or 4-8-membered ring are given. For the systematic derivation of all possible structures, graph theory may be introduced.

• Kyungpook Mathematical Journal
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• v.46 no.2
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• pp.153-167
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• 2006

We study permuting tri-derivations in ${\Gamma}$-rings and give an example.

• Communications of the Korean Mathematical Society
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• v.37 no.1
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• pp.57-64
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• 2022

In this paper, we show, among others, that if A is a Banach algebra satisfying a functional identity involving a b-generalized derivation F on A, under some mild conditions, is of the form F(x) = ax for all x ∈ R, where a ∈ Q_r, a right Martindale quotient ring of A.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.83-93
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• 2019

Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R and f($x_1,{\ldots},x_n$) a noncentral multilinear polynomial over C in n noncommuting variables. Denote by f(R) the set of all the evaluations of f($x_1,{\ldots},x_n$) on R. If d is a nonzero derivation of R and G a nonzero generalized derivation of R such that $$d(G(u)u){\in}Z(R)$$ for all $u{\in}f(R)$, then $f(x_1,{\ldots},x_n)^2$ is central-valued on R and there exists $b{\in}U$ such that G(x) = bx for all $x{\in}R$ with $d(b){\in}C$. As an application of this result, we investigate the commutator $[F(u)u,G(v)v]{\in}Z(R)$ for all $u,v{\in}f(R)$, where F and G are two nonzero generalized derivations of R.