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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Trees and valuation rings


Authors: Hans H. Brungs and Joachim Gräter
Journal: Trans. Amer. Math. Soc. 352 (2000), 3357-3379
MSC (2000): Primary 12E15, 12J20, 16K20
Published electronically: March 21, 2000
MathSciNet review: 1653339
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Abstract: A subring $B$ of a division algebra $D$ is called a valuation ring of $D$ if $x\in B$ or $x^{-1}\in B$ holds for all nonzero $x$ in $D$. The set $\mathcal{B}$ of all valuation rings of $D$ is a partially ordered set with respect to inclusion, having $D$ as its maximal element. As a graph $\mathcal{B}$ is a rooted tree (called the valuation tree of $D$), and in contrast to the commutative case, $\mathcal{B}$ may have finitely many but more than one vertices. This paper is mainly concerned with the question of whether each finite, rooted tree can be realized as a valuation tree of a division algebra $D$, and one main result here is a positive answer to this question where $D$ can be chosen as a quaternion division algebra over a commutative field.


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Additional Information

Hans H. Brungs
Affiliation: Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: hbrungs@vega.math.ualberta.ca

Joachim Gräter
Affiliation: Universität Potsdam, Institut für Mathematik, Postfach 601553, 14469 Potsdam, Germany
Email: graeter@rz.uni-potsdam.de

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02458-2
PII: S 0002-9947(00)02458-2
Keywords: Valuation rings, trees, division algebra
Received by editor(s): January 17, 1997
Published electronically: March 21, 2000
Additional Notes: The first author was supported in part by NSERC
Dedicated: In Memoriam Karl Mathiak
Article copyright: © Copyright 2000 American Mathematical Society