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Bamdad R Yahaghi - One of the best experts on this subject based on the ideXlab platform.
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an extension of a theorem of Kaplansky
Communications in Algebra, 2017Co-Authors: Heydar Radjavi, Bamdad R YahaghiAbstract:ABSTRACTA theorem of Kaplansky asserts that a semigroup of matrices with entries from a field whose members all have singleton spectra is triangularizable. Indeed, Kaplansky’s Theorem unifies well-known theorems of Kolchin and Levitzki on simultaneous triangularizability of semigroups of unipotent and nilpotent matrices, respectively. First, for a division ring D of characteristic zero whose center intersects its multiplicative commutator group in a finite group, we prove that the counterpart of Kolchin’s Theorem over D implies that of Kaplansky’s Theorem over D. Next, we note that this proof, when adjusted in the setting of fields, provides a new and simple proof of Kaplansky’s Theorem over fields of characteristic zero. We show that if Kaplansky’s Theorem holds over a division ring D, which is for instance the case over general fields, then a generalization of Kaplansky’s Theorem holds over D, and in particular over general fields.
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an extension of a theorem of Kaplansky
arXiv: Rings and Algebras, 2016Co-Authors: Heydar Radjavi, Bamdad R YahaghiAbstract:A theorem of Kaplansky asserts that a semigroup of matrices with entries from a field whose members all have singleton spectra is triangularizable. Indeed, Kaplansky's Theorem unifies well-known theorems of Kolchin and Levitzki on simultaneous triangularizability of semigroups of unipotent and nilpotent matrices, respectively. First, we present a new and simple proof of Kaplansky's Theorem over fields of characteristic zero. Next, we show that this proof can be adjusted to show that the counterpart of Kolchin's Theorem over division rings of characteristic zero implies that of Kaplansky's Theorem over such division rings. Also, we give a generalization of Kaplansky's Theorem over general fields. We show that this extension of Kaplansky's Theorem holds over a division ring $\Delta$ provided the counterpart of Kaplansky's Theorem holds over $\Delta$.
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a theorem of Kaplansky revisited
Linear Algebra and its Applications, 2015Co-Authors: Heydar Radjavi, Bamdad R YahaghiAbstract:Abstract We present a simple proof of a theorem due to Kaplansky which unifies theorems of Kolchin and Levitzki on triangularizability of semigroups of matrices. We also give two different extensions of the theorem. As a consequence, we prove the counterpart of Kolchin's Theorem for finite groups of unipotent matrices over division rings.
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a theorem of Kaplansky revisited
arXiv: Rings and Algebras, 2015Co-Authors: Heydar Radjavi, Bamdad R YahaghiAbstract:We present a new and simple proof of a theorem due to Kaplansky which unifies theorems of Kolchin and Levitzki on triangularizability of semigroups of matrices. We also give two different extensions of the theorem. As a consequence, we prove the counterpart of Kolchin's Theorem for finite groups of unipotent matrices over division rings. We also show that the counterpart of Kolchin's Theorem over division rings of characteristic zero implies that of Kaplansky's Theorem over such division rings.
Hwankoo Kim - One of the best experts on this subject based on the ideXlab platform.
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Kaplansky type theorems in graded integral domains
Bulletin of The Korean Mathematical Society, 2015Co-Authors: Gyuwhan Chang, Hwankoo KimAbstract:Abstract. It is well known that an integral domain D is a UFD if andonly if every nonzero prime ideal of D contains a nonzero principal prime.This is the so-called Kaplansky’s theorem. In this paper, we give this typeof characterizations of a graded PvMD (resp., G-GCD domain, GCDdomain, B´ezout domain, valuation domain, Krull domain, π-domain). 0. IntroductionThis is a continuation of our works on Kaplansky-type theorems [13, 20].It is well known that an integral domain D is a UFD if and only if everynonzero prime ideal of D contains a nonzero principal prime [19, Theorem 5].This is the so-called Kaplansky’s theorem. A generalization of this type oftheorems was first studied by Anderson and Zafrullah in [5], where they gaveseveral characterizations of this type for GCD domains, valuation domains, andPru¨fer domains. Also, characterizations for PvMDs and Krull domains weregiven in [15] and [6], respectively.Later, in [20], the second-named author gave a Kaplansky-type charac-terization of G-GCD domains and PvMDs and gave an ideal-wise version ofKaplansky-type theorems. This ideal-wise version is then used to give char-acterizations of UFDs, π-domains, and Krull domains. Let D be an integraldomain with quotient field K, X be an indeterminate over D, and D[X] bethe polynomial ring over D. A nonzero prime ideal Q of D[X] is called anupper to zero in D[X] if Q ∩ D = (0). Clearly, Q is an upper to zero in D[X]if and only if Q = fK[X] ∩ D[X] for some nonzero polynomial f ∈ D[X].For f ∈ D[X], let c(f) be the ideal of D generated by the coefficients of f.In [13], the first two authors of this paper studied an integral domain D suchthat every upper to zero in D[X] contains a prime (resp., primary) element,
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Kaplansky type theorems ii
Kyungpook Mathematical Journal, 2011Co-Authors: Gyuwhan Chang, Hwankoo KimAbstract:Let D be an integral domain with quotient field K, X be an indeterminate over D, and D[X] be the polynomial ring over D. A prime ideal Q of D[X] is called an upper to zero in D[X] if Q = fK[X] D[X] for some f D[X]. In this paper, we study integral domains D such that every upper to zero in D[X] contains a prime element (resp., a primary element, a t-invertible primary ideal, an invertible primary ideal).
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Kaplansky type theorems
Kyungpook Mathematical Journal, 2000Co-Authors: Hwankoo KimAbstract:We characterize G-GCD domains and PVMDs as Kaplansky-type theorems. We also give the unified ideal-wise version of a Kaplansky-type theorem. As a consequence, we recover well-known characterization of UFDs, , and Krull domains. Finally, we characterize PVMDs as a Nagata-type theorem.
Heydar Radjavi - One of the best experts on this subject based on the ideXlab platform.
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an extension of a theorem of Kaplansky
Communications in Algebra, 2017Co-Authors: Heydar Radjavi, Bamdad R YahaghiAbstract:ABSTRACTA theorem of Kaplansky asserts that a semigroup of matrices with entries from a field whose members all have singleton spectra is triangularizable. Indeed, Kaplansky’s Theorem unifies well-known theorems of Kolchin and Levitzki on simultaneous triangularizability of semigroups of unipotent and nilpotent matrices, respectively. First, for a division ring D of characteristic zero whose center intersects its multiplicative commutator group in a finite group, we prove that the counterpart of Kolchin’s Theorem over D implies that of Kaplansky’s Theorem over D. Next, we note that this proof, when adjusted in the setting of fields, provides a new and simple proof of Kaplansky’s Theorem over fields of characteristic zero. We show that if Kaplansky’s Theorem holds over a division ring D, which is for instance the case over general fields, then a generalization of Kaplansky’s Theorem holds over D, and in particular over general fields.
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an extension of a theorem of Kaplansky
arXiv: Rings and Algebras, 2016Co-Authors: Heydar Radjavi, Bamdad R YahaghiAbstract:A theorem of Kaplansky asserts that a semigroup of matrices with entries from a field whose members all have singleton spectra is triangularizable. Indeed, Kaplansky's Theorem unifies well-known theorems of Kolchin and Levitzki on simultaneous triangularizability of semigroups of unipotent and nilpotent matrices, respectively. First, we present a new and simple proof of Kaplansky's Theorem over fields of characteristic zero. Next, we show that this proof can be adjusted to show that the counterpart of Kolchin's Theorem over division rings of characteristic zero implies that of Kaplansky's Theorem over such division rings. Also, we give a generalization of Kaplansky's Theorem over general fields. We show that this extension of Kaplansky's Theorem holds over a division ring $\Delta$ provided the counterpart of Kaplansky's Theorem holds over $\Delta$.
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a theorem of Kaplansky revisited
Linear Algebra and its Applications, 2015Co-Authors: Heydar Radjavi, Bamdad R YahaghiAbstract:Abstract We present a simple proof of a theorem due to Kaplansky which unifies theorems of Kolchin and Levitzki on triangularizability of semigroups of matrices. We also give two different extensions of the theorem. As a consequence, we prove the counterpart of Kolchin's Theorem for finite groups of unipotent matrices over division rings.
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a theorem of Kaplansky revisited
arXiv: Rings and Algebras, 2015Co-Authors: Heydar Radjavi, Bamdad R YahaghiAbstract:We present a new and simple proof of a theorem due to Kaplansky which unifies theorems of Kolchin and Levitzki on triangularizability of semigroups of matrices. We also give two different extensions of the theorem. As a consequence, we prove the counterpart of Kolchin's Theorem for finite groups of unipotent matrices over division rings. We also show that the counterpart of Kolchin's Theorem over division rings of characteristic zero implies that of Kaplansky's Theorem over such division rings.
Kannan Soundararajan - One of the best experts on this subject based on the ideXlab platform.
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ramanujan s ternary quadratic form
Inventiones Mathematicae, 1997Co-Authors: Ken Ono, Kannan SoundararajanAbstract:do not seem to obey any simple law.” Following I. Kaplansky, we call a non-negative integer N eligible for a ternary form f(x, y, z) if there are no congruence conditions prohibiting f from representing N. By the classical theory of quadratic forms, it is well known that any given genus of positive definite ternary quadratic forms represents every eligible integer. Consequently if a genus consists of a single class with representative f(x, y, z), then f represents every eligible integer. In the case of Ramanujan’s form, this only implies that an eligible integer, one not of the form 4(16μ+ 6), is represented by φ1 or φ2.
Michael Puschnigg - One of the best experts on this subject based on the ideXlab platform.
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the kadison Kaplansky conjecture for word hyperbolic groups
Inventiones Mathematicae, 2002Co-Authors: Michael PuschniggAbstract:In this paper we prove the Kadison-Kaplansky idempotent conjecture for torsion-free word-hyperbolic groups. The conjecture asserts that the following equivalent statements hold for a torsion-free discrete group Γ: • The reduced group C∗-algebra C∗ r (Γ) contains no idempotents except 0 and 1. • The spectrum of every element of the reduced group C∗-algebra is connected. • The canonical trace on C∗ r (Γ) takes integer values on idempotents. The last assertion can be viewed as a statement about the pairing between the K-theory and the (local) cyclic cohomology of the group C∗-algebra. It is in this setting that we will prove the conjecture. Our proof is based on a partial analysis of the assembly maps in K-theory and local cyclic homology. We compare these assembly maps by means of an equivariant bivariant Chern-Connes character. Before going into details, we recall some previous work on the conjecture. The first progress was achieved by Pimsner and Voiculescu [PV] who proved the Kadison-Kaplansky conjecture for free groups as a consequence of their computation of the K-theory of the group C∗-algebra. Subsequently it was realized that more generally the Kadison-Kaplansky conjecture was a consequence of the Baum-Connes conjecture which gives a geometric description of K∗(C∗ r (Γ)) for any torsion-free discrete group. In fact the Baum-Connes conjecture states that the K-theoretic assembly map
