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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 |
|---|---|
| Autores principales: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Providence, Rhode Island :
American Mathematical Society,
2015.
|
| Colección: | Memoirs of the American Mathematical Society ;
no. 1128. |
| Temas: | |
| Acceso en línea: | Texto completo |
MARC
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| 100 | 1 | |a Dickmann, M. A., |d 1940- |e author. |1 https://id.oclc.org/worldcat/entity/E39PCjxHqMdFqTYkbM9dYK6fWP | |
| 245 | 1 | 0 | |a Faithfully quadratic rings / |c M. Dickmann, F. Miraglia. |
| 264 | 1 | |a Providence, Rhode Island : |b American Mathematical Society, |c 2015. | |
| 264 | 4 | |c ©2015 | |
| 300 | |a 1 online resource (xi, 129 pages) : |b illustrations | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
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| 490 | 1 | |a Memoirs of the American Mathematical Society, |x 0065-9266 ; |v volume 238, number 1128 | |
| 588 | 0 | |a Print version record. | |
| 504 | |a Includes bibliographical references (pages 121-123) and indexes. | ||
| 500 | |a "Volume 238, number 1129 (sixth of 6 numbers), November 2015." | ||
| 520 | |a 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. | ||
| 505 | 0 | |a Preface -- Basic concepts -- Rings and special groups -- The notion of T-faithfully quadratic ring. Some basic consequences -- Idempotents, products and T-isometry -- First-order axioms for quadratic faithfulness -- Rings with many units -- Transversality of representation in p-rings with bounded inversion -- Reduced f-rings -- Strictly representable rings -- Quadratic form theory over faithfully quadratic rings -- Bibliography -- Index of symbols -- Subject index. | |
| 590 | |a ProQuest Ebook Central |b Ebook Central Academic Complete | ||
| 650 | 0 | |a Forms, Quadratic. | |
| 650 | 0 | |a Commutative rings. | |
| 650 | 6 | |a Formes quadratiques. | |
| 650 | 6 | |a Anneaux commutatifs. | |
| 650 | 7 | |a Commutative rings |2 fast | |
| 650 | 7 | |a Forms, Quadratic |2 fast | |
| 650 | 7 | |a Quadratische Form |2 gnd | |
| 650 | 7 | |a Kommutativer Ring |2 gnd | |
| 700 | 1 | |a Miraglia, Francisco, |e author. | |
| 710 | 2 | |a American Mathematical Society, |e publisher. | |
| 758 | |i has work: |a Faithfully quadratic rings (Text) |1 https://id.oclc.org/worldcat/entity/E39PCGXYpYYWDdmrCM6f4jC4v3 |4 https://id.oclc.org/worldcat/ontology/hasWork | ||
| 776 | 0 | 8 | |i Print version: |a Dickmann, M.A., 1940- |t Faithfully quadratic rings. |d Providence, Rhode Island : American Mathematical Society, 2015 |z 9781470414689 |w (DLC) 2015027245 |w (OCoLC)915159430 |
| 830 | 0 | |a Memoirs of the American Mathematical Society ; |v no. 1128. | |
| 856 | 4 | 0 | |u https://ebookcentral.uam.elogim.com/lib/uam-ebooks/detail.action?docID=4832043 |z Texto completo |
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