On ordered division rings

Ismail M. Idris

Czechoslovak Mathematical Journal (2003)

  • Volume: 53, Issue: 1, page 69-76
  • ISSN: 0011-4642

Abstract

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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 → 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 the same 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.

How to cite

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Idris, Ismail M.. "On ordered division rings." Czechoslovak Mathematical Journal 53.1 (2003): 69-76. <http://eudml.org/doc/30759>.

@article{Idris2003,
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 $x \rightarrow x a^2$ 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 the same 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.},
author = {Idris, Ismail M.},
journal = {Czechoslovak Mathematical Journal},
keywords = {ordering; division ring; ordering; division ring},
language = {eng},
number = {1},
pages = {69-76},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On ordered division rings},
url = {http://eudml.org/doc/30759},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Idris, Ismail M.
TI - On ordered division rings
JO - Czechoslovak Mathematical Journal
PY - 2003
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 1
SP - 69
EP - 76
AB - 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 $x \rightarrow x a^2$ 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 the same 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.
LA - eng
KW - ordering; division ring; ordering; division ring
UR - http://eudml.org/doc/30759
ER -

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