On ordered division rings

Ismail M. Idris¹

¹ Department of Mathematics Faculty of Science Ain-Shams University Cairo, Egypt

Colloquium Mathematicum, Tome 88 (2001) no. 2, pp. 263-271.

Résumé

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.

Détail

DOI : 10.4064/cm88-2-8

Keywords: prestel introduced generalization notion ordering field which called semiordering prestels axioms semiordered field differ usual artin schreier postulates requiring only closedness domain positivity under mapsto non zero place requiring positive elements have positive product work study type ordering division ring actually behaves just commutative further bounded subring associated ordering valuation ring which preserved under conjugation associate semiordering natural valuation