Valuations and rank of ordered abelian groups
HTML articles powered by AMS MathViewer
- by Manish Kumar PDF
- Proc. Amer. Math. Soc. 133 (2005), 343-348 Request permission
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
- Shreeram Abhyankar, Ramification theoretic methods in algebraic geometry, Annals of Mathematics Studies, No. 43, Princeton University Press, Princeton, N.J., 1959. MR 0105416, DOI 10.1515/9781400881390
- Shreeram Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321–348. MR 82477, DOI 10.2307/2372519
- James R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR 0464128
- William J. Lewis and Jack Ohm, The ordering of $\textrm {Spec}$ $R$, Canadian J. Math. 28 (1976), no. 4, 820–835. MR 409428, DOI 10.4153/CJM-1976-079-2
Additional Information
- Manish Kumar
- Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
- Email: mkumar@math.purdue.edu
- 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
- © Copyright 2004
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 133 (2005), 343-348
- MSC (2000): Primary 12F10, 14H30, 20D06, 20E22
- DOI: https://doi.org/10.1090/S0002-9939-04-07692-0
- MathSciNet review: 2093053