Kumar, Manish Valuations and rank of ordered abelian groups. (English) Zbl 1125.13010 Proc. Am. Math. Soc. 133, No. 2, 343-348 (2005). Summary: 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 are derived and a necessary condition for an order type to be the rank of an ordered abelian group are discussed. These facts are translated to the spectrum of a valuation ring using some well-known results in valuation theory. Cited in 1 Document MSC: 13F30 Valuation rings 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 20F60 Ordered groups (group-theoretic aspects) Keywords:ordered abelian group; order type PDF BibTeX XML Cite \textit{M. Kumar}, Proc. Am. Math. Soc. 133, No. 2, 343--348 (2005; Zbl 1125.13010) Full Text: DOI References: [1] Shreeram Abhyankar, Ramification theoretic methods in algebraic geometry, Annals of Mathematics Studies, no. 43, Princeton University Press, Princeton, N.J., 1959. · Zbl 0093.04501 [2] Shreeram Abhyankar, On the valuations centered in a local domain, Amer. J. Math. 78 (1956), 321 – 348. · Zbl 0074.26301 · doi:10.2307/2372519 · doi.org [3] James R. Munkres, Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. · Zbl 0306.54001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.