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 ↦ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-2-8,
author = {Ismail M. Idris},
title = {On ordered division rings},
journal = {Colloquium Mathematicae},
volume = {89},
year = {2001},
pages = {263-271},
zbl = {1005.06010},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-2-8}
}
Ismail M. Idris. On ordered division rings. Colloquium Mathematicae, Tome 89 (2001) pp. 263-271. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm88-2-8/