Weak ∗-Orderings on ∗-Fields

✍ Scribed by K.H. Leung

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 815 KB
• Volume: 156
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1993.1067

✦ Synopsis

Analogous to the notion of natural valuations of ordered fields, we introduce the notion of order *-valuations for any Baer ordered *-fields. When the Bear ordered division rings are finite dimensional over their centers, we show that their order *-valuations are nontrivial. Using this, we study a new generalization of *-orderings, namely, weak *-orderings. Unlike *-orderings, weak *-orderings do exist in Bear ordered *-fields odd dimensional over their centers. Moreover, we prove that if the involution is of the first kind, these -fields must be either commutative fields or standard quaternion algebras. Whereas in case the involution is of the second kind, the dimension of these *-fields over their centers must be odd. This strong result also implies that the restriction of weak -ordering on any commutative subfield consisting of symmetric elements only is in fact an ordering (not just a semiordering) is these ()-fields. 1993 Academic Press. Inc.