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Valuations and rank of ordered abelian groups
Author:
Manish Kumar
Journal:
Proc. Amer. Math. Soc. 133 (2005), 343-348
MSC (2000):
Primary 12F10, 14H30, 20D06, 20E22
Published electronically:
August 25, 2004
MathSciNet review:
2093053
Full-text PDF Free Access
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Abstract: It is shown that there exists an ordered abelian group that has no smallest positive element and that has no sequence of nonzero elements converging to zero. Some formulae for the rank of ordered abelian groups have been derived and a necessary condition for an order type to be rank of an ordered abelian group has been discussed. These facts have been translated to the spectrum of a valuation ring using some well-known results in valuation theory.
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Shreeram
Abhyankar, Ramification theoretic methods in algebraic
geometry, Annals of Mathematics Studies, no. 43, Princeton University
Press, Princeton, N.J., 1959. MR 0105416
(21 #4158)
- [Ab2]
Shreeram
Abhyankar, On the valuations centered in a local domain, Amer.
J. Math. 78 (1956), 321–348. MR 0082477
(18,556b)
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James
R. Munkres, Topology: a first course, Prentice-Hall Inc.,
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820–835. MR 0409428
(53 #13183)
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- S. S. Abhyankar, Ramification Theoretic Methods in Algebraic Geometry, Princeton University Press, 1959. MR 0105416 (21:4158)
- [Ab2]
- S. S. Abhyankar, On valuations centered in the local domain, American Journal of Mathematics, 78 (1956), 321-348. MR 0082477 (18:556b)
- [Mun]
- J. R. Munkres, A First Course in Topology, Prentice Hall Inc., 1975. MR 0464128 (57:4063)
- [LOh]
- W. J. Lewis, J. Ohm, The ordering of
 , Canadian Journal of Mathematics 28 (1976), no. 4, 820-835. MR 0409428 (53:13183)
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Additional Information
Manish Kumar
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email:
mkumar@math.purdue.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-04-07692-0
PII:
S 0002-9939(04)07692-0
Keywords:
Ordered abelian groups,
spectrum of valuation rings
Received by editor(s):
April 25, 2003
Published electronically:
August 25, 2004
Additional Notes:
The author thanks Prof. Shreeram S. Abhyankar for the motivation and support provided in developing the theory and in verifying the proof.
Communicated by:
Bernd Ulrich
Article copyright:
© Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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