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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Value functions and Dubrovin valuation rings on simple algebras
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by Mauricio A. Ferreira and Adrian R. Wadsworth PDF
Trans. Amer. Math. Soc. 368 (2016), 1681-1734 Request permission

Abstract:

In this paper we prove relationships between two generalizations of commutative valuation theory for noncommutative central simple algebras: (1) Dubrovin valuation rings; and (2) the value functions called gauges introduced by Tignol and Wadsworth. We show that if $v$ is a valuation on a field $F$ with associated valuation ring $V$ and $v$ is defectless in a central simple $F$-algebra $A$, and $C$ is a subring of $A$, then the following are equivalent: (a) $C$ is the gauge ring of some minimal $v$-gauge on $A$, i.e., a gauge with the minimal number of simple components of $C/J(C)$; (b) $C$ is integral over $V$ with $C = B_1 \cap \ldots \cap B_\xi$ where each $B_i$ is a Dubrovin valuation ring of $A$ with center $V$, and the $B_i$ satisfy Gräter’s Intersection Property. Along the way we prove the existence of minimal gauges whenever possible and we show how gauges on simple algebras are built from gauges on central simple algebras.
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Additional Information
  • Mauricio A. Ferreira
  • Affiliation: Departamento de Ciências Exatas, Universidade Estadual de Feira de Santana, Avenida Transnordestina, S/N, Novo Horizonte, Feira de Santana, Bahia 44036-900 Brazil
  • Email: maferreira@uefs.br
  • Adrian R. Wadsworth
  • Affiliation: Department of Mathematics 0112, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112
  • Email: arwadsworth@ucsd.edu
  • Received by editor(s): July 20, 2013
  • Received by editor(s) in revised form: December 31, 2013
  • Published electronically: June 17, 2015
  • Additional Notes:

    Some of the results in this paper are based on the first author’s doctoral dissertation written under the supervision of the second author and Antonio José Engler. The first author was partially supported by FAPESP, Brazil (Grant 06/00157-3)

    during his graduate studies. This author would like to thank the second author and UCSD for their hospitality during his visit in 2009.


    Some of the research for this paper was carried out during the second author’s visit to the University of Campinas, Brazil, during July-August, 2010, which was made possible by a grant from FAPESP, Brazil. The second author would like to thank Professor Antonio José Engler and the first author for their hospitality during that visit.
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 1681-1734
  • MSC (2010): Primary 16W60; Secondary 16K20, 16W70, 16L30
  • DOI: https://doi.org/10.1090/tran/6363
  • MathSciNet review: 3449223