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Abstract: The main purpose of this paper is to prove the following result: Let n > 1 be a fixed integer, let R be a n!-torsion free semiprime ring, and let f : R -> R be an additive mapping satisfying the relation [f (x), x]n = [[. . . [[f (x), x], x], . . .], x] = 0 for all x = R. In this case [f (x), x] = 0 is fulfilled for all x = R. Since any semisimple Banach algebra (for example, C algebra) is semiprime, this purely algebraic result might be of some interest from functional analysis point of view.
Keywords: mathematics, algebra, semiprime rings, derivation
Published in DKUM: 27.06.2017; Views: 1266; Downloads: 547
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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: 375
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Abstract: We prove in this note the following result. Let ▫$n>1$▫ be an integer and let ▫$R$▫ be an ▫$n!$▫-torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping ▫$D : R \to R$▫ such that ▫$D(x^n)=\Sigma_{j^n}=1^{x^{n-j}}D(x)x^{j-1}$▫ is fulfilled for all ▫$ x \in R$▫. In this case, ▫$D$▫ is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, ▫$R$▫ will represent an associative ring with center ▫$Z(R)$▫. Given an integer ▫$n > 1$▫, a ring ▫$R$▫ is said to be ▫$n$▫-torsion-free if for ▫$x \in R$▫, ▫$nx=0$▫ implies that ▫$x=0$▫. Recall that a ring ▫$R$▫ is prime if for ▫$ a,b \in R$▫, ▫$aRb=(0)$▫ implies that either ▫$a=0$▫ or ▫$b=0$▫, and is semiprime in case ▫$aRa=(0)$▫ implies that ▫$a=0$▫. An additive mapping ▫$D:R \to R$▫ is called a derivation if ▫$D(xy)=D(x)y+xD(y)$▫ holds for all pairs ▫$x,y \in R$▫ and is called a Jordan derivation in case ▫$D(x^2)=D(x)x+xD(x)$▫ is fulfilled for all ▫$x \in R$▫. Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein (1957) asserts that any Jordan derivation on a prime ring with characteristic different from two is a derivation. A brief proof of Herstein's result can be found in 1988 by Brešar and Vukman. Cusack (1975) generalized Herstein's result to ▫$2$▫-torsion-free semiprime rings (see also Brešar (1988) for an alternative proof). For some other results concerning derivations on prime and semiprime rings, we refer to [2, 7, 8, 9, 10].
Keywords: mathematics, associative rings an algebras, derivations, semiprime rings
Published in DKUM: 14.06.2017; Views: 1254; Downloads: 366
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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: 552
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Abstract: V doktorski disertaciji si bomo v uvodu ogledali nekaj osnovnih pojmov, definicij ter pomembnejših rezultatov s področja algebre. Obravnavali bomo funkcionalne enačbe, ki so v zvezi z odvajanji, centralizatorji ter sorodnimi preslikavami na prakolobarjih, polprakolobarjih in standardnih operatorskih algebrah. Na tem področju že vrsto let delujejo tudi slovenski matematiki, ki so s svojimi rezultati pomembno vplivali na razvoj tega področja. Že v osemdesetih letih sta bila močno dejavna na tem področju J. Vukman, M. Brešar, sledili pa so B. Zalar, B. Hvala, v novejšem času pa M. Fošner, I. Kosi-Ulbl, D. Benkovič, D. Eremita, A. Fošner, N. Peršin ter N. Širovnik. Osnovno sredstvo pri reševanju funkcionalnih enačb, ki bodo predstavljene v disertaciji, je teorija funkcijskih identitet, ki jo je leta 2000 v cite{87} predstavil M. Brešar. Leta 2007 pa so jo M. Brešar, M. A. Chebotar in W. S. Martindale III tudi podrobneje predstavili v knjigi cite{MB4}. Teorija funkcijskih identitiet bo v disertaciji predstavljena skupaj s polinomskimi identitietami ter d-prostimi množicami.
Keywords: Aditivna preslikava, linearen operator, odvajanje, jordansko odvajanje, jordansko trojno odvajanje, centralizator, funkcionalna enačba, standardna operatorska algebra, prakolobar, polprakolobar, Banachov prostor, involucija.
Published in DKUM: 21.10.2016; Views: 2278; Downloads: 147
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