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Faithfully quadratic rings /
In this monograph we extend the classical algebraic theory of quadratic forms over fields to diagonal quadratic forms with invertible entries over broad classes of commutative, unitary rings where -1 is not a sum of squares and 2 is invertible. We accomplish this by: (1) Extending the classical noti...
| Clasificación: | Libro Electrónico |
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| Autores principales: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Providence, Rhode Island :
American Mathematical Society,
2015.
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| Colección: | Memoirs of the American Mathematical Society ;
no. 1128. |
| Temas: | |
| Acceso en línea: | Texto completo |
| Sumario: | In this monograph we extend the classical algebraic theory of quadratic forms over fields to diagonal quadratic forms with invertible entries over broad classes of commutative, unitary rings where -1 is not a sum of squares and 2 is invertible. We accomplish this by: (1) Extending the classical notion of matrix isometry of forms to a suitable notion of T-isometry, where T is a preorder of the given ring, A, or T = A². (2) Introducing in this context three axioms expressing simple properties of (value) representation of elements of the ring by quadratic forms, well-known to hold in the field case. Under these axioms we prove that the ring-theoretic approach based on T-isometry coincides with the formal approach formulated in terms of reduced special groups. This guarantees, for rings verifying these axioms, the validity of a number of important structural properties, notably the Arason-Pfister Hauptsatz, Milnor's mod 2 Witt ring conjecture, Marshall's signature conjecture, uniform upper bounds for the Pfister index of quadratic forms, a local-global Sylvester inertia law, etc. We call (T)-faithfully quadratic rings verifying these axioms. A significant part of the monograph is devoted to prove quadratic faithfulness of certain outstanding (classes of) rings; among them, rings with many units satisfying a mild additional requirement, reduced f-rings (herein rings of continuous real-valued functions), and strictly representable rings. Obviously, T-quadratic faithfulness depends on both the ring and the preorder T. We isolate a property of preorders defined solely in terms of the real spectrum of a given ring -- that we baptise unit-reflecting preorders -- which, for an extensive class of preordered rings, (A, T), turns out to be equivalent to the T-quadratic faithfulness of A. We show, e.g., that all preorders on the ring of continuous real-valued functions on a compact Hausdorff are unit-reflecting; we also give examples where this property fails. |
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| Notas: | "Volume 238, number 1129 (sixth of 6 numbers), November 2015." |
| Descripción Física: | 1 online resource (xi, 129 pages) : illustrations |
| Bibliografía: | Includes bibliographical references (pages 121-123) and indexes. |
| ISBN: | 9781470426293 1470426293 1470414686 9781470414689 |
| ISSN: | 0065-9266 ; |