• 대한수학회보
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• 제44권4호
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• pp.789-794
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• 2007

Let R be a 3-torsion free semiprime ring and let I be a nonzero two-sided ideal of R. Suppose that there exists a permuting 3-derivation ${\Delta}:R{\times}R{\times}R{\rightarrow}R$ such that the trace is centralizing on I. Then the trace of ${\Delta}$ is commuting on I. In particular, if R is a 3!-torsion free prime ring and ${\Delta}$ is nonzero under the same condition, then R is commutative.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제14권4호
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• pp.271-278
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• 2007

Let R be a 2-torsion free semiprime ring. Suppose that there exists a 4-permuting 4-derivation ${\Delta}:R{\times}R{\times}R{\times}R{\rightarrow}R$ such that the trace is centralizing on R. Then the trace of ${\Delta}$ is commuting on R. In particular, if R is a 3!-torsion free prime ring and ${\Delta}$ is nonzero under the same condition, then R is commutative.

• 대한수학회보
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• 제32권2호
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• pp.367-372
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• 1995

In 1955 Singer and Wermer [12] proved that every bounded derivation on a commutative Banach algebra maps into its radical. They conjectured that the continuity of the derivation in their theorm can be removed. In 1988 Thomas [13] proved their conjecture ; Every derivation on a commutative Banach algebra maps into its radical. For noncommutative versions, in 1984 B. Yood [15] proved that the continuous derivations on Banach algebras satisfing [D(a),b] $\in$ Rad(A) for all a, b $\in$ A have the radical range, where [a,b] will be denote the commutator ab-ba. In 1990 M.Bresar and J.Vukman [1] have generlized Yood's result, that is, the continuous linear Jordan derivation on Banach algebra that satisfies [D(a),a] $\in$ Rad(A) for all a $\in$ A has the radical range. In next year Mathieu and Murphy [5] proved that every bounded centralizing derivation on Banach algebras has its image in the radical. Mathieu and Runde [6] removed the boundedness of that.

• Kyungpook Mathematical Journal
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• 제62권1호
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• pp.43-55
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• 2022

Let R be a *-ring and n ≥ 1 be an integer. The objective of this paper is to introduce the notion of n-skew centralizing maps on *-rings, and investigate the impact of these maps. In particular, we describe the structure of prime rings with involution '*' such that _*[x, d(x)]_n ∈ Z(R) for all x ∈ R (for n = 1, 2), where d : R → R is a nonzero derivation of R. Among other related results, we also provide two examples to prove that the assumed restrictions on our main results are not superfluous.

• 충청수학회지
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• 제22권3호
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• pp.451-458
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• 2009

Let $n{\geq}2$ be a fixed positive integer and let R be a noncommutative n!-torsion free semiprime ring. Suppose that there exists a symmetric n-derivation $\Delta$ : $R^{n}{\rightarrow}R$ such that the trace of $\Delta$ is centralizing on R. Then the trace is commuting on R. If R is a n!-torsion free prime ring and $\Delta{\neq}0$ under the same condition. Then R is commutative.

• 대한수학회보
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• 제52권4호
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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 $L^{\infty}_0({\omega})^*/L^1({\omega})$.

• 충청수학회지
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• 제24권3호
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• pp.543-551
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• 2011

In this paper, we consider the functional inequalities with approximately centralizing derivations on noncommutative Banach algebras, and investigate the problem that functions satisfying the functional inequalities mentioned above map into the radical.

• 대한수학회논문집
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• 제26권3호
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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.

• Journal of applied mathematics & informatics
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• 제6권1호
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• pp.239-246
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• 1999

The purpose of this paper is to prove the following result: Let R be a 2-torsion free noncommutative semi-prime ring and D:RlongrightarrowR a derivation. Suppose that $[[D(\chi),\chi],\chi]\in$ Z(R) holds for all $\chi \in R$. Then D is commuting on R.

• 대한수학회논문집
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• 제34권4호
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• pp.1099-1104
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• 2019

In [1], Ali and Dar proved the ${\ast}$-version of classical theorem due to Posner [15, Theorem] with involution of the second kind. The main objective of this paper is to improve the above mentioned result without the condition of the second kind involution. Moreover, a related result has been discussed.