Semisimple Modules & Rings and the Wedderburn Structure Theorem

Existence

10.23943/princeton/9780691166902.003.0016 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Division Algebra ◽

Quadratic Forms ◽

Structure Theorem ◽

Quadratic Extension ◽

Quadratic Space ◽

Division Algebras ◽

Separable Extension ◽

Valued Fields ◽

Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

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On Lattice-Ordered Rees Matrix Γ-Semigroups

Annals of the Alexandru Ioan Cuza University - Mathematics ◽

10.2478/v10157-012-0033-8 ◽

2013 ◽

Vol 59 (1) ◽

pp. 209-218 ◽

Cited By ~ 2

Author(s):

Kostaq Hila ◽

Edmond Pisha

Keyword(s):

Structure Theorem ◽

Lattice Of Congruences

Abstract The purpose of this paper is to introduce and give some properties of l-Rees matrix Γ-semigroups. Generalizing the results given by Guowei and Ping, concerning the congruences and lattice of congruences on regular Rees matrix Γ-semigroups, the structure theorem of l-congruences lattice of l - Γ-semigroup M = μº(G : I; L; Γe) is given, from which it follows that this l-congruences lattice is distributive.

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A structure theorem for polyharmonic maps between Riemannian manifolds

Journal of Differential Equations ◽

10.1016/j.jde.2020.11.046 ◽

2021 ◽

Vol 273 ◽

pp. 14-39

Author(s):

Volker Branding

Keyword(s):

Riemannian Manifolds ◽

Structure Theorem

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Quasi-ideal Ehresmann transversals: The spined product structure

Open Mathematics ◽

10.1515/math-2020-0071 ◽

2021 ◽

Vol 19 (1) ◽

pp. 77-86

Author(s):

Xiangjun Kong ◽

Pei Wang ◽

Jian Tang

Keyword(s):

Structure Theorem ◽

Product Structure ◽

Abundant Semigroup ◽

Spined Product ◽

Equivalent Conditions

Abstract In any U-abundant semigroup with an Ehresmann transversal, two significant components R and L are introduced in this paper and described by Green’s ∼ \sim -relations. Some interesting properties associated with R and L are explored and some equivalent conditions for the Ehresmann transversal to be a quasi-ideal are acquired. Finally, a spined product structure theorem is established for a U-abundant semigroup with a quasi-ideal Ehresmann transversal by means of R and L.

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Infinite approximate subgroups of soluble Lie groups

Mathematische Annalen ◽

10.1007/s00208-021-02258-8 ◽

2021 ◽

Author(s):

Simon Machado

Keyword(s):

Lie Groups ◽

Structure Theorem ◽

Quasi Crystals

AbstractWe study infinite approximate subgroups of soluble Lie groups. We show that approximate subgroups are close, in a sense to be defined, to genuine connected subgroups. Building upon this result we prove a structure theorem for approximate lattices in soluble Lie groups. This extends to soluble Lie groups a theorem about quasi-crystals due to Yves Meyer.

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