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Abstract: In this article, we prove the following result. Let n ≥ 3 be some fixed integer and let R be a prime ring with ≠ + − char R n 1 !2n 2 ( ) ( ) . Suppose there exists an additive mapping D : R → R satisfying the relation 2n−2 D ( x n ) = ( n − 2 ∑ i = 0 ( n − 2 i ) x i D ( x 2 ) x n − 2 − i ) + ( 2 n − 2 − 1 ) ( D ( x ) x n − 1 + x n − 1 D ( x ) ) + n − 2 ∑ i = 1 ( i ∑ k = 2 ( 2 k − 1 − 1 ) ( n − k − 2 i − k ) + n − 1 − i ∑ k = 2 ( 2 k − 1 − 1 ) ( n − k − 2 n − i − k − 1 ) ) x i D ( x ) x n − 1 − i for all x ∈ R. In this case, D is a derivation. This result is related to a classical result of Herstein, which states that any Jordan derivation on a prime ring with char(R) ≠ 2 is a derivation.
Keywords: prime ring, semiprime ring, derivation, Jordan derivation, functional equation
Published in DKUM: 18.07.2025; Views: 0; Downloads: 7
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Abstract: Let ▫$A$▫ be an algebra over a field ▫$F$▫ with ▫$\mathrm{char} (F) \ne 2$▫. If ▫$A$▫ is generated as an algebra by ▫$[[A,A],[A,A]]$▫, then for every skew-symmetric bilinear map ▫$\Phi:A \times A \to X$▫, where ▫$X$▫ is an arbitrary vector space over ▫$F$▫, the condition that ▫$\Phi(x^2,x)=0$▫ for all ▫$x \in A$▫ implies that ▫$\Phi(xy,z) +\Phi(zx,y) + \Phi(yz,x)=0$▫ for all ▫$x,y,z \in A$▫. This is applicable to the question of whether ▫$A$▫ is zero Lie product determined, and is also used in proving that a Jordan homomorphism from ▫$A$▫ onto a semiprime algebra ▫$B$▫ is the sum of a homomorphism and an antihomomorphism.
Keywords: bilinear map, zero Lie product determined algebra, derivation, Jordan derivation, Jordan homomorphism, functional identity
Published in DKUM: 18.08.2023; Views: 421; Downloads: 46
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Abstract: Let R be an associative ring. An element ▫$a\in R$▫ is said to be dependent on a mapping ▫$F:R\to R$▫ in case ▫$F(x)a=ax$▫ holds for all ▫$x\in R$▫. In this paper, elements dependent on certain mappings on prime and semiprime rings are investigated. We prove, for example, that in case we have a semiprime ring R, there are no nonzero elements which are dependent on the mapping ▫$\alpha + \beta$▫, where an ▫$\alpha$▫ and ▫$\beta$▫ are automorphisms of R
Keywords: mathematics, algebra, rings, algebras, derivation, Jordan derivation, left centralizer, right centralizer, additive mapping, dependent elements
Published in DKUM: 14.06.2017; Views: 1090; Downloads: 384
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Abstract: In calculus, an indefinite integral of a function f is a differentiable function F whose derivative is equal to f. The main goal of the paper is to generalize this notion of the indefinite integral from the ring of real functions to any ring. We also investigate basic properties of such generalized integrals and compare them to the well-known properties of indefinite integrals of real functions.
Keywords: ring, integration, Jordan integration, derivation, Jordan derivation
Published in DKUM: 10.05.2017; Views: 1151; Downloads: 214
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Abstract: The purpose of this paper is to prove the following result. Let ▫$ m\geq\ge 1$▫, ▫$n \geq\ge 1$▫ be some fixed integers with ▫$m \ne n$▫, and let R be a prime ring with ▫$char (R) \ne 2mn (m+n) l, \vert m-n l, \vert$▫. Suppose there exists a nonzero additive mapping ▫$D : R \to R$▫ satisfying the relation ▫$(m + n)D(x^2) = 2mD(x)x + 2nxD(x)$▫ for all ▫$x \in R ((m,n)-Jordan derivation)$▫. If either ▫$char(R) = 0$▫ or ▫$char(R) \geq 3$▫ then D is a derivation and R is commutative.
Keywords: prime rings, derivation, Jordan derivation, commutativity
Published in DKUM: 31.03.2017; Views: 1258; Downloads: 593
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