1. Commuting graphs and extremal centralizersGregor Dolinar, Aleksandr Èmilevič Guterman, Bojan Kuzma, Polona Oblak, 2014, original scientific article Abstract: We determine the conditions for matrix centralizers which can guarantee the connectedness of the commuting graph for the full matrix algebra ▫$M_n(\mathbb{F})$▫ over an arbitrary field ▫$\mathbb{F}$▫. It is known that if ▫$\mathbb{F}$▫ is an algebraically closed field and ▫$n \ge 3$▫, then the diameter of the commuting graph of ▫$M_n(\mathbb{F})$▫ is always equal to four. We construct a concrete example showing that if ▫$\mathbb{F}$▫ is not algebraically closed, then the commuting graph of ▫$M_n(\mathbb{F})$▫ can be connected with the diameter at least five.
Keywords: commuting graph, matrix ring, centralizer
Published in RUP: 31.12.2021; Views: 798; Downloads: 23
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2. On maximal distances in a commuting graphGregor Dolinar, Bojan Kuzma, Polona Oblak, 2012, original scientific article Abstract: It is shown that matrices over algebraically closed fields that are farthest apart in the commuting graph must be non-derogatory. Rank-one matrices and diagonalizable matrices are also characterized in terms of the commuting graph.
Keywords: matematika, linearna algebra, teorija grafov, komutirajoči grafi, matrična algebra, algebraično zaprt obseg, centralizator, razdalja v grafih, mathematics, linear algebra, graph theory, commuting graph, matrix algebra, algebraically closed field, centralizer, distance in graphs
Published in RUP: 03.04.2017; Views: 2314; Downloads: 256
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3. A note on generalized (m,n)-Jordan centralizersAjda Fošner, 2013, original scientific article Abstract: The aim of this paper is to define generalized ▫$(m, n)$▫-Jordan centralizers and to prove that on a prime ring with nonzero center and ▫${\rm char}(R) \ne 6mn(m+n)(m+2n)$▫ every generalized ▫$(m, n)$▫-Jordan centralizer is a two-sided centralizer.
Keywords: mathematics, prime ring, semiprime ring, left (right) centralizer, left (right) Jordan centralizer, (m, n)-Jordan centralizer, generalized (m, n)-Jordan centralizer
Published in RUP: 15.10.2013; Views: 3542; Downloads: 170
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4. On generalized Jordan triple ([alpha], [beta]) [sup] [ast]-derivations and related mappingsShakir Ali, Ajda Fošner, Maja Fošner, Mohammad Salahuddin Khan, 2013, original scientific article Abstract: Let ▫$R$▫ be a 2-torsion free semiprime ▫$\ast$▫-ring and let ▫$\alpha, \beta$▫ be surjective endomorphisms of ▫$R$▫. The aim of the paper is to show that every generalized Jordan triple ▫$(\alpha, \beta)^\ast$▫-derivation on ▫$R$▫ is a generalized Jordan ▫$(\alpha, \beta)^\ast$▫-derivation. This result makes it possible to prove that every generalized Jordan triple ▫$(\alpha, \beta)^\ast$▫-derivation on a semisimple ▫$H^\ast$▫-algebra is a generalized Jordan ▫$(\alpha, \beta)^\ast$▫-derivation. Finally, we prove that every Jordan triple left ▫$\alpha^\ast$▫-centralizer on a 2-torsion free semiprime ring is a Jordan left ▫$\alpha^\ast$▫-centralizer.
Keywords: mathematics, algebra, semiprime ▫$\ast$▫-ring, ▫$H^\ast$▫-algebra, Jordan triple ▫$(\alpha, \beta)^\ast$▫-derivation, generalized Jordan triple ▫$(\alpha, \beta)^\ast$▫-derivation, Jordan triple left ▫$\alpha^\ast$▫-centralizer
Published in RUP: 15.10.2013; Views: 4424; Downloads: 77
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