Trees and valuation rings |
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| Authors: | Hans H. Brungs Joachim Grä ter |
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| Affiliation: | Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 ; Universität Potsdam, Institut für Mathematik, Postfach 601553, 14469 Potsdam, Germany |
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| 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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| Keywords: | Valuation rings trees division algebra |
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