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Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Valuations and rank of ordered abelian groups

Author(s): Manish Kumar
Journal: Proc. Amer. Math. Soc. 133 (2005), 343-348.
MSC (2000): Primary 12F10, 14H30, 20D06, 20E22
Posted: August 25, 2004
MathSciNet review: 2093053
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Abstract | References | Similar articles | Additional information

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.

References:

[Ab1]
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 $Spec R$, 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: 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
Posted: 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
Copyright of article: Copyright 2004, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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