Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [BG] H. H. Brungs, J. Gräter, Valuation Rings in Finite-Dimensional Division Algebras, J. Algebra 120 (1989), 90-99. MR 90a:16005
  • [BS] S. I. Borewicz, I. R. Safarevic, Zahlentheorie, Birkhäuser, 1966. MR 33:4000
  • [C] P. M. Cohn, On extending valuations in division algebras, Studia Sci. Math. Hungar. 16 (1981), 65-70. MR 84f:16020
  • [E] O. Endler, Valuation Theory, Springer, 1972. MR 50:9847
  • [G] J. Gräter, Über Bewertungen endlich dimensionaler Divisionsalgebren, Results in Math. 7 (1984), 54-57. MR 86d:12017
  • [K] I. Kersten, Brauergruppen von Körpern, Aspekte der Mathematik 6, Vieweg, 1990. MR 93e:16026
  • [M] K. Mathiak, Valuations of Skew Fields and Projective Hjelmslev Spaces, LNM 1175, Springer, 1986. MR 87g:16002
  • [Mo] P. Morandi, The Henselization of a Valued Division Algebra, J. Algebra 122 (1989), 232-243. MR 90h:12007
  • [O] O. Ore, Theory of Graphs, AMS Colloquium Publications Vol. XXXVIII, 1962. MR 27:740
  • [P] R. S. Pierce, Associative Algebras, Grad. Texts in Math. 88, Springer, 1982. MR 84c:16001
  • [S] W. Scharlau, Quadratic and Hermitian Forms, Springer, 1985. MR 86k:11022
  • [ZS] O. Zariski, P. Samuel, Commutative Algebra II, Grad. Texts in Math. 29, Springer, 1992. MR 22:11006 (original ed.)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 12E15, 12J20, 16K20

Retrieve articles in all journals with MSC (2000): 12E15, 12J20, 16K20

Additional Information

Hans H. Brungs
Affiliation: Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G1

Joachim Gräter
Affiliation: Universität Potsdam, Institut für Mathematik, Postfach 601553, 14469 Potsdam, Germany

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

American Mathematical Society