• 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 $({\sigma},{\tau})-derivation$.

• 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.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.

• 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.17 no.2
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• pp.225-232
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• 2001

Let R be a prime ring with characteristic not two. U a (${\sigma},{\tau}$)-left Lie ideal of R and d : R$\rightarrow$R a non-zero derivation. The purpose of this paper is to invesitigate identities satisfied on prime rings. We prove the following results: (1) [d(R),a]=0$\Leftrightarrow$d([R,a])=0. (2) if $(R,a)_{{\sigma},{\tau}}$=0 then $a{\in}Z$. (3) if $(R,a)_{{\sigma},{\tau}}{\subset}C_{{\sigma},{\tau}}$ then $a{\in}Z$. (4) if $(U,a){\subset}Z$ then $a^2{\in}Z\;or\;{\sigma}(u)+{\tau}(u){\in}Z$, for all $u{\in}U$. (5) if $(U,R)_{{\sigma},{\tau}}{\subset}C_{{\sigma},{\tau}}$ then $U{\subset}Z$.

• 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.

• 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.

• 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.

• Kyungpook Mathematical Journal
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• v.60 no.2
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• pp.415-421
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• 2020

In the proof of Theorem 3 on p.253 in [4], both right and left distributivity are assumed simultaneously which makes the proof invalid. We give a corrected proof for this theorem by introducing an extension of Lemma 2.2 in [2].

• Communications of the Korean Mathematical Society
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• v.32 no.4
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• pp.827-833
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• 2017

In this paper, we define a set including of all $f_a$ with $a{\in}R$ generalized derivations of R and is denoted by $f_R$. It is proved that (i) the mapping $g:L(R){\rightarrow}f_R$ given by g (a) = f-a for all $a{\in}R$ is a Lie epimorphism with kernel $N_{{\sigma},{\tau}}$ ; (ii) if R is a semiprime ring and ${\sigma}$ is an epimorphism of R, the mapping $h:f_R{\rightarrow}I(R)$ given by $h(f_a)=i_{{\sigma}(-a)}$ is a Lie epimorphism with kernel $l(f_R)$ ; (iii) if $f_R$ is a prime Lie ring and A, B are Lie ideals of R, then $[f_A,f_B]=(0)$ implies that either $f_A=(0)$ or $f_B=(0)$.