• 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.

• Communications of the Korean Mathematical Society
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• v.33 no.1
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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.

• Communications of the Korean Mathematical Society
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• v.32 no.3
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• pp.535-542
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• 2017

Let R be an associative ring 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 $D:R{\rightarrow}R$ is called a generalized (${\alpha},{\beta}$)-derivation of R associated with an (${\alpha},{\beta}$)-derivation d if $D(xy)=D(x){\alpha}(y)+{\beta}(x)d(y)$ is fulfilled for all $x,y{\in}R$. In this note, we intend to generalize a theorem of Vukman [5], and a theorem of Daif and El-Sayiad [2].

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.27-38
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• 2014

The notion of generalized (regular) (${\alpha},\;{\beta}$)-derivations of a BCI-algebra is introduced, some useful examples are discussed, and related properties are investigated. The condition for a generalized (${\alpha},\;{\beta}$)-derivation to be regular is provided. The concepts of a generalized F-invariant (${\alpha},\;{\beta}$)-derivation and ${\alpha}$-ideal are introduced, and their relations are discussed. Moreover, some results on regular generalized (${\alpha},\;{\beta}$)-derivations are proved.

• 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.

• Journal of the Korean Mathematical Society
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• v.48 no.4
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• pp.823-835
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• 2011

We investigate the continuity of (${\alpha},{\beta}$)-derivations on B(X) or $C^*$-algebras. We give some sufficient conditions on which (${\alpha},{\beta}$)-derivations on B(X) are continuous and show that each (${\alpha},{\beta}$)-derivation from a unital $C^*$-algebra into its a Banach module is continuous when and ${\alpha}$ ${\beta}$ are continuous at zero. As an application, we also study the ultraweak continuity of (${\alpha},{\beta}$)-derivations on von Neumann algebras.

• Communications of the Korean Mathematical Society
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• v.31 no.1
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• pp.101-113
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• 2016

In this article, we considered the stability of the following (${\alpha}$, ${\beta}$, ${\gamma}$)-derivation $${\alpha}D[x,y]={\beta}[D(x),y]+{\gamma}[x,D(y)]$$ and homomorphisms associated to the quadratic type functional equation $$f(kx+y)+f(kx+{\sigma}(y))=2kg(x)+2g(y),\;x,y{\in}A$$, where ${\sigma}$ is an involution of the Lie $C^*$-algebra A and k is a fixed positive integer. The Hyers-Ulam stability on unbounded domains is also studied. Applications of the results for the asymptotic behavior of the generalized quadratic functional equation are provided.

• Korean Journal of Plant Resources
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• v.20 no.6
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• pp.548-552
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• 2007

This research is the basic research to develop new anti-inflammatory medicine by feeding Scutellariae Radix extract to lipopolysaccharide(LPS) exposed rats, and analyzed it's effect on inflammatory response by LPS derivation. As a result, Plasma interleukin-$1\beta(IL-1\beta)$ and Plasma interleukin-6(IL-6) concentration showed the highest point at 5h after LPS injection, and in this time, the concentration of $IL-1\beta$ and IL-6 in the Scutellariae Radix extract groups at 200mg/kg and 300mg/kg showed lower values than that of control group. Plasma tumor necrosis $factor-\alpha(TNF-\alpha)$ concentration after LPS injection showed the highest point at 2h and showed similar level till at 5h. $TNF-\alpha$ concentration at 2h after LPS injection showed the low value only in the Scutellariae Radix extract 300mg/kg group compared to others, and in 5h, the all Scutellariae Radix extract groups showed lower value than that of the control group. Plasma interleukin-10(IL-10) concentration increased at 2h after LPS injection and reached the highest at 5h. After LPS injection the IL-10 concentration at 2h, the Scutellariae Radix extract injection group at 300mg/kg showed higher value than that of the others, and in 5h after LPS injection, Scutellariae Radix extract 200mg and 300mg groups showed higher value than the control group. Concluding from the above results, in inflammatory response by LPS derivation, the Scutellariae Radix gives positive effect.

• Journal of the Korean Mathematical Society
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• v.35 no.3
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• pp.559-574
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• 1998

Let the Gel'fand triple (E)$_{\beta}$/ ⊂ ( $L^2$) ⊂ (E)＊$_{\beta}$/ be the framework of white noise distribution theory constructed by Kon-dratiev and Streit. A new class of continuous multiplicative products on (E)$_{\beta}$/ is introduced and associated continuous derivations on (E)$_{\beta}$/ are discussed. Algebraic characterizations of first order differential operators on (E)$_{\beta}$/ are proved. Some applications are also discussed. $\beta$/ are proved. Some applications are also discussed.

• Journal of Applied and Pure Mathematics
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• v.4 no.1_2
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• pp.1-7
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• 2022

A mapping T : R → R (not necessarily additive) is called multiplicative left 𝛼-centralizer if T(xy) = T(x)𝛼(y) for all x, y ∈ R. A mapping F : R → R (not necessarily additive) is called multiplicative (generalized)(𝛼, 𝛽)-derivation if there exists a map (neither necessarily additive nor derivation) f : R → R such that F(xy) = F(x)𝛼(y) + 𝛽(x)f(y) for all x, y ∈ R, where 𝛼 and 𝛽 are automorphisms on R. The main purpose of this paper is to study some algebraic identities with multiplicative (generalized) (𝛼, 𝛽)-derivations and multiplicative left 𝛼-centralizer on the left ideal of a prime ring R.