# Article

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Keywords:
ordering; division ring
Summary:
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 $x \rightarrow x a^2$ 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.
References:
[1] A.  Prestel: Lectures on Formally Real Fields. Lecture Notes in Math. 1093. Springer Verlag, , 1984. MR 0769847
[2] T. Szele: On ordered skew fields. Proc. Amer. Math. Soc. 3 (1952), 410–413. DOI 10.1090/S0002-9939-1952-0047017-7 | MR 0047017 | Zbl 0047.03104

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