On ordered division rings

Volume 88 / 2001

Ismail M. Idris Colloquium Mathematicum 88 (2001), 263-271 MSC: Primary 06F25, 16W10. DOI: 10.4064/cm88-2-8


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 $x\mapsto xa^2$ for non-zero $a$, in place of requiring that positive elements have a positive product. Our aim in this work is to study this type of ordering in the case of a division ring. We show that it actually behaves just as in the commutative case. Further, we show that the bounded subring associated with that ordering is a valuation ring which is preserved under conjugation, so one can associate with the semiordering a natural valuation.


  • Ismail M. IdrisDepartment of Mathematics
    Faculty of Science
    Ain-Shams University
    Cairo, Egypt

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