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150824t20152015riua ob 001 0 eng d |
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|a UAB
|b eng
|e rda
|e pn
|c UAB
|d OCLCO
|d UIU
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|d OCLCF
|d YDX
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|d OCLCA
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|a 958353687
|a 1086459747
|a 1262672961
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| 020 |
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|a 9781470426293
|q (online)
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|a 1470426293
|q (online)
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|a 1470414686
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|a 9781470414689
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|z 9781470414689
|q (print)
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|z 1470414686
|q (print)
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3 |
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|a 9781470414689
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| 035 |
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|a (OCoLC)920024541
|z (OCoLC)958353687
|z (OCoLC)1086459747
|z (OCoLC)1262672961
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| 050 |
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4 |
|a QA243
|b .D476 2015
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| 049 |
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|a HCDD
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| 100 |
1 |
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|a Dickmann, M. A.,
|d 1940-
|e author.
|1 https://id.oclc.org/worldcat/entity/E39PCjxHqMdFqTYkbM9dYK6fWP
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| 245 |
1 |
0 |
|a Faithfully quadratic rings /
|c M. Dickmann, F. Miraglia.
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| 264 |
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1 |
|a Providence, Rhode Island :
|b American Mathematical Society,
|c 2015.
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| 264 |
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4 |
|c ©2015
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| 300 |
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|a 1 online resource (xi, 129 pages) :
|b illustrations
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| 336 |
|
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|a text
|b txt
|2 rdacontent
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| 337 |
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|a computer
|b c
|2 rdamedia
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| 338 |
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|a online resource
|b cr
|2 rdacarrier
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| 490 |
1 |
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|a Memoirs of the American Mathematical Society,
|x 0065-9266 ;
|v volume 238, number 1128
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| 588 |
0 |
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|a Print version record.
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| 504 |
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|a Includes bibliographical references (pages 121-123) and indexes.
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| 500 |
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|a "Volume 238, number 1129 (sixth of 6 numbers), November 2015."
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| 520 |
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|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.
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| 505 |
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|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.
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| 650 |
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|a Forms, Quadratic.
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| 650 |
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|a Commutative rings.
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| 650 |
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7 |
|a Commutative rings
|2 fast
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| 650 |
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7 |
|a Forms, Quadratic
|2 fast
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| 650 |
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7 |
|a Quadratische Form
|2 gnd
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| 650 |
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7 |
|a Kommutativer Ring
|2 gnd
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| 700 |
1 |
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|a Miraglia, Francisco,
|e author.
|1 https://id.oclc.org/worldcat/entity/E39PCjtMrk8BQDh7JxQDc3QMdP
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| 710 |
2 |
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|a American Mathematical Society,
|e publisher.
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| 758 |
|
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|i has work:
|a Faithfully quadratic rings (Text)
|1 https://id.oclc.org/worldcat/entity/E39PCGXYpYYWDdmrCM6f4jC4v3
|4 https://id.oclc.org/worldcat/ontology/hasWork
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| 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.proquest.com/lib/holycrosscollege-ebooks/detail.action?docID=4832043
|y Click for online access
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| 903 |
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|a EBC-AC
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| 994 |
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|a 92
|b HCD
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