• The Pure and Applied Mathematics
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• v.25 no.2
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• pp.135-147
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• 2018

Lu et al. [27] defined derivations on fuzzy Banach spaces and fuzzy Lie Banach spaces and proved the Hyers-Ulam stability of derivations on fuzzy Banach spaces and fuzzy Lie Banach spaces. It is easy to show that the definitions of derivations on fuzzy Banach spaces and fuzzy Lie Banach spaces are wrong and so the results of [27] are wrong. Moreover, there are a lot of seroius problems in the statements and the proofs of the results in Sections 2 and 3. In this paper, we correct the definitions of biderivations on fuzzy Banach algebras and fuzzy Lie Banach algebras and the statements of the results in [27], and prove the corrected theorems.

• Communications of the Korean Mathematical Society
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• v.39 no.2
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• pp.353-361
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• 2024

We consider 𝒩 to be a 3-prime field and 𝒫 to be a prime ideal of 𝒩. In this paper, we study the commutativity of the quotient near-ring 𝒩/𝒫 with left multipliers and derivations satisfying certain identities on 𝒫, generalizing some well-known results in the literature. Furthermore, an example is given to illustrate the necessity of our hypotheses.

• 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 the Chungcheong Mathematical Society
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• v.32 no.2
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• pp.195-205
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• 2019

We consider additive mappings similar to derivations on Banach ${\ast}$-algebras and we will first study the conditions for such additive mappings on Banach ${\ast}$-algebras. Then we prove some theorems concerning approximate linear mappings of derivation-type on Banach ${\ast}$-algebras. As an application, approximate linear mappings of derivation-type on $C^{\ast}$-algebra are characterized.

• Bulletin of the Korean Mathematical Society
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• v.48 no.4
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• pp.853-871
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• 2011

In this paper, we prove the Hyers-Ulam-Rassias stability of the following additive functional inequality: ${\parallel}f(2x)+f(2y)+2f(z){\parallel}\;{\leq}\;{\parallel}2f(x+y+z){\parallel}$ We investigate homomorphisms in proper $CQ^*$-algebras and derivations on proper $CQ^*$-algebras associated with the additive functional inequality (0.1).

• Communications of the Korean Mathematical Society
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• v.22 no.1
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• pp.41-51
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• 2007

Let R be a ring with left identity e and suitably-restricted additive torsion, and Z(R) its center. Let H : $R{\times}R{\times}R{\rightarrow}R$ be a symmetric 3-additive mapping, and let h be the trace of H. In this paper we show that (i) if for each $x{\in}R$, $$n=<<\cdots,\;x>,\;\cdots,x>{\in}Z(R)$$ with $n\geq1$ fixed, then h is commuting on R. Moreover, h is of the form $$h(x)=\lambda_0x^3+\lambda_1(x)x^2+\lambda_2(x)x+\lambda_3(x)\;for\;all\;x{\in}R$$, where $\lambda_0\;{\in}\;Z(R)$, $\lambda_1\;:\;R{\rightarrow}R$ is an additive commuting mapping, $\lambda_2\;:\;R{\rightarrow}R$ is the commuting trace of a bi-additive mapping and the mapping $\lambda_3\;:\;R{\rightarrow}Z(R)$ is the trace of a symmetric 3-additive mapping; (ii) for each $x{\in}R$, either $n=0\;or\;<n,\;x^m>=0$ with $n\geq0,\;m\geq1$ fixed, then h = 0 on R, where denotes the product yx+xy and Z(R) is the center of R. We also present the conditions which implies commutativity in rings with identity as motivated by the above result.

• Honam Mathematical Journal
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• v.31 no.3
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• pp.407-419
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• 2009

The simple non-associative algebra $N(e^{A_S},q,n,t)_k$ and its simple sub-algebras are defined in the papers [1], [3], [4], [5], [6], [12]. We define the non-associative algebra $\overline{WN_{(g_n,\mathfrak{U}),m,s_B}}$ and its antisymmetrized algebra $\overline{WN_{(g_n,\mathfrak{U}),m,s_B}}$. We also prove that the algebras are simple in this work. There are various papers on finding all the derivations of an associative algebra, a Lie algebra, and a non-associative algebra (see [3], [5], [6], [9], [12], [14], [15]). We also find all the derivations $Der_{anti}(WN(e^{{\pm}x^r},0,2)_B^-)$ of te antisymmetrized algebra $WN(e^{{\pm}x^r}0,2)_B^-$ and every derivation of the algebra is outer in this paper.

• Korean Journal of Mathematics
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• v.27 no.4
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• pp.969-976
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• 2019

Let ℬ(ℋ) be the algebra of all bounded linear operators on a complex Hilbert space ℋ and 𝒨 ⊆ ℬ(ℋ) be a von Neumann algebra without central abelian projections. Let ξ be a non-zero scalar. In this paper, it is proved that a mapping φ : 𝒨 → ℬ(ℋ) satisfies φ([A, B]^ξ_⁎)= [φ(A), B]^ξ_⁎+[A, φ(B)]^ξ_⁎ for all A, B ∈ 𝒨 if and only if φ is an additive ⁎-derivation and φ(ξA) = ξφ(A) for all A ∈ 𝒨.

• Communications of the Korean Mathematical Society
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• v.28 no.4
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• pp.767-782
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• 2013

We deal with the Hyers-Ulam stability problem of linear mappings from a vector space into a Banach one with respect to the following functional equation: $$f\(\frac{-x+y}{3}\)+f\(\frac{x-3z}{3}\)+f\(\frac{3x-y+3z}{3}\)=f(x)$$. We then combine this equation with other ones and establish the Hyers-Ulam stability of several kinds of linear mappings, among which the algebra (*-) homomorphisms, the derivations, the multipliers and others. We thus repair and improve some previous assertions in the literature.

• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.1-12
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• 2015

Let R be a ring and $D:R^n{\longrightarrow}R$ be n-additive mapping. A map $d:R{\longrightarrow}R$ is said to be the trace of D if $d(x)=D(x,x,{\ldots}x)$ for all $x{\in}R$. Suppose that ${\alpha},{\beta}$ are endomorphisms of R. For any $a,b{\in}R$, let < a, b > $_{({\alpha},{\beta})}=a{\alpha}(b)+{\beta}(b)a$. In the present paper under certain suitable torsion restrictions it is shown that D = 0 if R satisfies either < d(x), $x^m$ > $_{({\alpha},{\beta})}=0$, for all $x{\in}R$ or ${\ll}$ d(x), x > $_{({\alpha},{\beta})}$, $x^m$ > $_{({\alpha},{\beta})}=0$, for all $x{\in}R$. Further, if < d(x), x > ${\in}Z(R)$, the center of R, for all $x{\in}R$ or < d(x)x - xd(x), x >= 0, for all $x{\in}R$, then it is proved that d is commuting on R. Some more related results are also obtained for additive mapping on R.