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Valuations and rank of ordered abelian groups. (English) Zbl 1125.13010
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.

MSC:
13F30 Valuation rings
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
20F60 Ordered groups (group-theoretic aspects)
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[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
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