• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1863-1871
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• 2013

For a ring R with an automorphism ${\sigma}$, an n-additive mapping ${\Delta}:R{\times}R{\times}{\cdots}{\times}R{\rightarrow}R$ is called a skew n-derivation with respect to ${\sigma}$ if it is always a ${\sigma}$-derivation of R for each argument. Namely, if n - 1 of the arguments are fixed, then ${\Delta}$ is a ${\sigma}$-derivation on the remaining argument. In this short note, from Bre$\check{s}$ar Theorems, we prove that a skew n-derivation ($n{\geq}3$) on a semiprime ring R must map into the center of R.

• Bulletin of the Korean Mathematical Society
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• v.45 no.4
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• pp.729-737
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• 2008

In this paper we characterize multi-Jensen functions f : $V^n\;{\rightarrow}\;W$, where n is a positive integer, V, W are commutative groups and V is uniquely divisible by 2. Moreover, under the assumption that f : $\mathbb{R}\;{\rightarrow}\;\mathbb{R}$ is Borel measurable, we obtain representation of f (respectively, f, g, h : $\mathbb{R}\;{\rightarrow}\;\mathbb{R}$) such that the Jensen difference $$2f\;\(\frac{x\;+\;y}{2}\)\;-\;f(x)\;-\;f(y)$$ (respectively, the Pexider difference $$2f\;\(\frac{x\;+\;y}{2}\)\;-\;g(x)\;-\;h(y))$$ takes values in a countable subgroup of $\mathbb{R}$.

• Journal of the Chungcheong Mathematical Society
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• v.19 no.2
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• pp.189-196
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• 2006

The generalized Hyers-Ulam stability problems of the cubic functional equation f(x + y + z) + f(x + y - z) + 2f(x - y) + 4f(y) = f(x - y + z) + f(x - y - z) +2f(x + y) + 2f(y + z) + 2f(y - z) shall be treated under the approximately odd condition and the behavior of the cubic mappings and the additive mappings shall be investigated. The generalized Hyers-Ulam stability problem for functional equations had been posed by Th.M. Rassias and J. Tabor [7] in 1992.

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.69-76
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• 2012

Let R be a ring, and ${\alpha}$ be an endomorphism of R. An additive mapping H : R ${\rightarrow}$ R is called a left ${\alpha}$-multiplier (centralizer) if H(xy) = H(x)${\alpha}$(y) holds for all x,y $\in$ R. In this paper, we shall investigate the commutativity of prime and semiprime rings admitting left ${\alpha}$-multiplier satisfying any one of the properties: (i) H([x,y])-[x,y] = 0, (ii) H([x,y])+[x,y] = 0, (iii) $H(x{\circ}y)-x{\circ}y=0$, (iv) $H(x{\circ}y)+x{\circ}y=0$, (v) H(xy) = xy, (vi) H(xy) = yx, (vii) $H(x^2)=x^2$, (viii) $H(x^2)=-x^2$ for all x, y in some appropriate subset of R.

• Journal of the Korean Mathematical Society
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• v.45 no.1
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• pp.249-271
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• 2008

We prove the Hyers-Ulam stability of linear d-isometries in linear d-normed Banach modules over a unital $C^*-algebra$ and of linear isometries in Banach modules over a unital $C^*-algebra$. The main purpose of this paper is to investigate d-isometric $C^*-algebra$ isomor-phisms between linear d-normed $C^*-algebras$ and isometric $C^*-algebra$ isomorphisms between $C^*-algebras$, and d-isometric Poisson $C^*-algebra$ isomorphisms between linear d-normed Poisson $C^*-algebras$ and isometric Poisson $C^*-algebra$ isomorphisms between Poisson $C^*-algebras$. We moreover prove the Hyers-Ulam stability of their d-isometric homomorphisms and of their isometric homomorphisms.

• The Pure and Applied Mathematics
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• v.23 no.3
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• pp.237-246
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• 2016

Ebadian et al. proved the Hyers-Ulam stability of bimultipliers and Jordan bimultipliers in C*-ternary algebras by using the fixed point method. Under the conditions in the main theorems for bimultipliers, we can show that the related mappings must be zero. Moreover, there are some mathematical errors in the statements and the proofs of the results. In this paper, we correct the statements and the proofs of the results, and prove the corrected theorems by using the direct method.

• Communications of the Korean Mathematical Society
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• v.36 no.2
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• pp.229-238
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• 2021

Let R be a ring with characteristic different from 2. An additive mapping F : R → R is called a generalized derivation on R if there exists a derivation d : R → R such that F(xy) = F(x)y + xd(y) holds for all x, y ∈ R. In the present paper, we show that if R is a prime ring satisfying certain identities involving a generalized derivation F associated with a derivation d, then R becomes commutative and in some cases d comes out to be zero (i.e., F becomes a left centralizer). We provide some counter examples to justify that the restrictions imposed in the hypotheses of our theorems are not superfluous.

• Journal of Applied and Pure Mathematics
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• v.4 no.3_4
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• pp.99-105
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• 2022

The objective of this research paper is to prove that an additive mapping T from a semiprime ring R to itself will be centralizer having a suitable torsion restriction on R if it satisfy any one of the following algebraic equations (a) 2T(xⁿyⁿxⁿ) = T(xⁿ)yⁿxⁿ + xⁿyⁿT(xⁿ) (b) 3T(xⁿyⁿxⁿ) = T(xⁿ)yⁿxⁿ+xⁿT(yⁿ)xⁿ+xⁿyⁿT(xⁿ) for every x, y ∈ R. Further, few extensions of these results are also presented in the framework of *-ring.

• Korean Journal of Mathematics
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• v.31 no.1
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• pp.95-107
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• 2023

Consider ℛ to be an associative prime ring and 𝒦 to be a nonzero dense ideal of ℛ. A mapping (need not be additive) ℱ : ℛ → 𝒬_mr associated with derivation d : ℛ → ℛ is called a multiplicative b-generalized derivation if ℱ(αδ) = ℱ(α)δ +bαd(δ) holds for all α, δ ∈ ℛ and for any fixed (0 ≠)b ∈ 𝒬_s ⊆ 𝒬_mr. In this manuscript, we study the commutativity of prime rings when the map b-generalized derivation satisfies the strong commutativity preserving condition and moreover, we investigate the commutativity of prime rings that admit multiplicative b-generalized derivation, which improves many results in the literature.

• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.709-717
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• 2022

The objective of this research is to prove that an additive mapping T : R → R is a left as well as right centralizer on R if it satisfies any one of the following identities: (i) T(xⁿyⁿ + yⁿxⁿ) = T(xⁿ)yⁿ + yⁿT(xⁿ) (ii) 2T(xⁿyⁿ) = T(xⁿ)yⁿ + yⁿT(xⁿ) for each x, y ∈ R, where n ≥ 1 is a fixed integer and R is any n!-torsion free semiprime ring. In addition, we talk over above identities in the setting of *-ring(ring with involution).