• Kyungpook Mathematical Journal
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• v.62 no.1
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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.

• Honam Mathematical Journal
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• v.35 no.4
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• pp.625-637
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• 2013

We investigate anti-centralizing and skew-centralizing mappings involving generalized derivations and derivations on prime and semiprime rings. We also obtain some range inclusion results for generalized linear derivations and linear derivations on Banach algebras by applying the algebraic techniques. Some results in this note are to improve the ones in [22].

• Journal of the Chungcheong Mathematical Society
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• v.24 no.3
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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.

• The Pure and Applied Mathematics
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• v.14 no.4
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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.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.3
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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.

• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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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.

• Communications of the Korean Mathematical Society
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• v.36 no.4
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• pp.641-650
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• 2021

The notions of skew-commuting/commuting/semi-commuting/skew-centralizing/semi-centralizing mappings play an important role in ring theory. ${\mathfrak{C}}^*$-algebras with these properties have been studied considerably less and the existing results are motivating the researchers. This article elaborates the structure of prime rings and ${\mathfrak{C}}^*$-algebras satisfying certain functional identities involving automorphisms.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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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 $L^{\infty}_0({\omega})^*/L^1({\omega})$.

• Bulletin of the Korean Mathematical Society
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• v.32 no.2
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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.

• Communications of the Korean Mathematical Society
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• v.20 no.1
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• pp.23-34
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• 2005

Let R be a ring with left identity. Let G : $R{\times}R{\to}R$ be a symmetric biadditive mapping and g the trace of G. Let ${\alpha}\;:\;R{\to}R$ be an endomorphism and ${\beta}\;:\;R{\to}R$ an epimorphism. In this paper we show the following: (i) Let R be 2-torsion-free. If g is (${\alpha},{\beta}$)-skew-commuting on R, then we have G = 0. (ii) If g is (${\beta},{\beta}$)-skew-centralizing on R, then g is (${\beta},{\beta}$)-commuting on R. (iii) Let $n{\ge}2$. Let R be (n+1)!-torsion-free. If g is n-(${\alpha},{\beta}$)-skew-commuting on R, then we have G = 0. (iv) Let R be 6-torsion-free. If g is 2-(${\alpha},{\beta}$)-commuting on R, then g is (${\alpha},{\beta}$)-commuting on R.