• Journal of Applied and Pure Mathematics
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• v.6 no.1_2
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• pp.13-20
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• 2024

Let R be a ring with involution. In the present paper, we characterize biadditive mappings which satisfies some functional identities related to symmetric Jordan (𝛼, 1)^*-biderivation of prime rings with involution. In particular, we prove that on a 2-torsion free prime ring with involution, every symmetric Jordan triple (𝛼, 1)^*-biderivation is a symmetric Jordan (𝛼, 1)^*-biderivation.

• Journal of Applied and Pure Mathematics
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• v.6 no.3_4
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• pp.155-165
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• 2024

Let R be a semi-prime ring. Let [δ₁, D₁] and [δ₂, D₂] be two generalized symmetric reverse biderivations of R with associated reverse biderivations D₁ and D₂. The main aim of the present paper is to establish conditions of orthogonality for symmetric reverse biderivations and symmetric generalized reverse biderivations in R.

• Kyungpook Mathematical Journal
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• v.51 no.3
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• pp.301-309
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• 2011

In this paper we dene the notions of left *-bimultiplier, *-bimultiplier and generalized *-biderivation, and to prove that if a semiprime *-ring admits a left *-bimultiplier M, then M maps R ${\times}$ R into Z(R). In Section 3, we discuss the applications of theory of *-bimultipliers. Further, it was shown that if a semiprime *-ring R admits a symmetric generalized *-biderivation G : R ${\times}$ R ${\rightarrow}$ R with an associated nonzero symmetric *-biderivation R ${\times}$ R ${\rightarrow}$ R, then G maps R ${\times}$ R into Z(R). As an application, we establish corresponding results in the setting of $C^*$-algebra.

• The Pure and Applied Mathematics
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• v.24 no.2
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• pp.99-107
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• 2017

Shokri et al. [14] proved the Hyers-Ulam stability of bihomomorphisms and biderivations by using the direct method. It is easy to show that the definition of biderivations on normed 3-Lie algebras is meaningless and so the results of [14] are meaningless. In this paper, we correct the definition of biderivations and the statements of the results in [14], and prove the corrected theorems.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.385-397
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• 2022

Let (M, [ , ]) be a finite dimensional Malcev algebra over an algebraically closed field 𝔽 of characteristic 0. We first prove that, (M, [ , ]) (with [M, M] ≠ 0) is simple if and only if ind(M) = 1 (i.e., M admits a unique (up to a scalar multiple) invariant scalar product). Further, we characterize the form of skew-symmetric biderivations on simple Malcev algebras. In particular, we prove that the simple seven dimensional non-Lie Malcev algebra has no nontrivial skew-symmetric biderivation.

• 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.