• East Asian mathematical journal
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• v.23 no.2
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• pp.197-206
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• 2007

The purpose of this paper is to show some results of M. Bresar and J. Vukman concerning orthogonal derivations of semiprime rings in [3] for (${\sigma},\;{\tau}$)-derivations and generalized (${\sigma},\;{\tau}$)-derivations in $\Gamma$-rings.

• East Asian mathematical journal
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• v.24 no.4
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• pp.389-398
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• 2008

In this paper, some results of [6] concerning orthogonal ($\sigma$, $\tau$)-derivations and generalized ($\sigma$, $\tau$)-derivations are generalized for a nonzero ideal of a semiprime ring.

• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.441-448
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• 2012

Let R be a prime ring with center Z and characteristic different from two, U a nonzero Lie ideal of R such that $u^2{\in}U$ for all $u{\in}U$ and $d$ be a nonzero (${\sigma}$, ${\tau}$)-derivation of R. We prove the following results: (i) If $[d(u),u]_{{\sigma},{\tau}}$ = 0 or $[d(u),u]_{{\sigma},{\tau}}{\in}C_{{\sigma},{\tau}}$ for all $u{\in}U$, then $U{\subseteq}Z$. (ii) If $a{\in}R$ and $[d(u),a]_{{\sigma},{\tau}}$ = 0 for all $u{\in}U$, then $U{\subseteq}Z$ or $a{\in}Z$. (iii) If $d([u,v])={\pm}[u,v]_{{\sigma},{\tau}}$ for all $u{\in}U$, then $U{\subseteq}Z$.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1121-1129
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• 2010

In the present paper, we extend some well known results concerning derivations of prime rings to generalized derivations for ($\sigma,\tau$)-Lie ideals.

• Communications of the Korean Mathematical Society
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• v.26 no.3
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• pp.445-454
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• 2011

In this paper, we extend some well known results concerning generalized derivations of prime rings to a generalized (${\sigma}$, ${\tau}$)-derivation.

• Bulletin of the Korean Mathematical Society
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• v.43 no.1
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• pp.9-16
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• 2006

The purpose of this paper is to define the symmetric $bi-(\sigma,\;\tau)$ derivations on prime and semi prime Gamma rings and to prove some results concerning symmetric $bi-(\sigma,\;\tau)$derivations on prime and semi prime Gamma rings.

• Kyungpook Mathematical Journal
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• v.50 no.3
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• pp.379-387
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• 2010

In this paper, we present some results concerning orthogonal generalized (${\sigma},{\tau}$)-derivations in semiprime near-rings. These results are a generalization of result of Bresar and Vukman, which are related to a theorem of Posner for the product of two derivations in prime rings.

• 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 $({\sigma},{\tau})-derivation$.

• The Pure and Applied Mathematics
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• v.10 no.4
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• pp.207-215
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• 2003

Let R be a prime ring with characteristic different from two and let $\theta,\varphi,\sigma,\tau$ be the automorphisms of R. Let d : $R{\rightarrow}R$ be a nonzero ($\theta,\varphi$)-derivation. We prove the following results: (i) if $a{\in}R$ and [d(R), a]$_{{\theta}o{\sigma},{\varphi}o{\tau}}$=0, then $\sigma(a)\;+\;\tau(a)\;\in\;Z$, the center of R, (ii) if $d([R,a]_{\sigma,\;\tau)\;=\;0,\;then\;\sigma(a)\;+\;\tau(a)\;\in\;Z$, (iii) if $[ad(x),\;x]_{\sigma,\;\tau}\;=\;0;for\;all\;x\;\in\;RE$, then a = 0 or R is commutative.

• The Pure and Applied Mathematics
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• v.13 no.3 s.33
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• pp.189-195
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• 2006

Let R be a prime ring with characteristics not 2 and ${\sigma},\;{\tau},\;{\alpha},\;{\beta}$ be auto-morphisms of R. Suppose that $d_1$ is a (${\sigma},\;{\tau}$)-derivation and $d_2$ is a (${\alpha},\;{\beta}$)-derivation on R such that $d_{2}{\alpha}\;=\;{\alpha}d_2,\;d_2{\beta}\;=\;{\beta}d_2$. In this note it is shown that; (1) If $d_1d_2$(R) = 0 then $d_1$ = 0 or $d_2$ = 0. (2) If [$d_1(R),d_2(R)$] = 0 then R is commutative. (3) If($d_1(R),d_2(R)$) = 0 then R is commutative. (4) If $[d_1(R),d_2(R)]_{\sigma,\tau}$ = 0 then R is commutative.