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

DOI:
https://doi.org/10.1090/S0002-9947-00-02458-2

Published electronically:
March 21, 2000

MathSciNet review:
1653339

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Abstract | References | Similar Articles | Additional Information

Abstract: A subring of a division algebra is called a valuation ring of if or holds for all nonzero in . The set of all valuation rings of is a partially ordered set with respect to inclusion, having as its maximal element. As a graph is a rooted tree (called the valuation tree of ), and in contrast to the commutative case, 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 , and one main result here is a positive answer to this question where 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:
https://doi.org/10.1090/S0002-9947-00-02458-2

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