Abstract
Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel's axioms for a semiordered field differ from the usual (Artin-Schreier) postulates in requiring only the closedness of the domain of positivity under x → xa 2 for nonzero a, instead of requiring that positive elements have a positive product. In this work, this type of ordering is studied in the case of a division ring. It is shown that it actually behaves the same as in the commutative case. Further, it is shown that the bounded subring associated with that ordering is a valuation ring which is preserved under conjugation, so one can associate a natural valuation to a semiordering.
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References
A.Prestel: Lectures on Formally Real Fields. Lecture Notes in Math. 1093. Springer Verlag, 1984.
T.Szele: On ordered skew fields. Proc. Amer. Math. Soc. 3 (1952), 410–413.
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Idris, I.M. On Ordered Division Rings. Czechoslovak Mathematical Journal 53, 69–76 (2003). https://doi.org/10.1023/A:1022971424461
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DOI: https://doi.org/10.1023/A:1022971424461