On ordered division rings

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 (Formula presented.) xa2 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.