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On ordered division rings. (English) Zbl 1014.06017

Summary: Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel’s axioms for semiordered fields 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 in the same way 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.

MSC:

06F25 Ordered rings, algebras, modules
16W80 Topological and ordered rings and modules
12E15 Skew fields, division rings
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References:

[1] A. Prestel: Lectures on Formally Real Fields. Lecture Notes in Math. 1093. Springer Verlag, , 1984.
[2] T. Szele: On ordered skew fields. Proc. Amer. Math. Soc. 3 (1952), 410-413. · Zbl 0047.03104 · doi:10.2307/2031894
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