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Strong ordering of *-fields. (English) Zbl 0624.06024
This paper is the third in a series of three papers (so far) in which the author discusses orderings on *-fields [the others are J. Algebra 46, 207–219 (1977; Zbl 0359.12023) and Trans. Am. Math. Soc. 262, 219–243 (1980; Zbl 0482.12009)]. A field (i.e. division ring) with involution, *, is Baer-ordered in case it contains a subset called the domain of positivity satisfying five axioms, each of which is easily verified in the classical case, $${\mathbb R}^+$$ in $${\mathbb R}$$, $${\mathbb C}$$, or $${\mathbb H}$$. The strong ordering introduced in this paper requires, in addition, that the product of positive elements be positive.
I quote from the introduction, ‘...this new ordering cannot apply to some *-fields orderable in the sense of Baer. But when it does apply it is easier to handle and has richer consequences. It is the purpose of this paper to present some of these consequences, especially as they contrast with what is known about Baer’s ordering.’
The paper is by no means self-contained. However this is a blessing, the ample references to related work will allow the reader to verify and expand on particular points, while the lack of detail makes for an accessible read. This is especially useful for those who, like the reviewer, are interested in these results primarily for the insights they may yield into the structure of related systems such as Baer *-rings and coordinatized ortholattices, (cf. the introduction to the second paper cited).
Reviewer: M.Roddy

MSC:
 12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) 06F25 Ordered rings, algebras, modules 16Kxx Division rings and semisimple Artin rings 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 16D90 Module categories in associative algebras 12J15 Ordered fields
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References:
 [1] Baer, R, Über nicht-archimedisch geordnate Körper, S.-B. heidelberger akad. wiss., 8, 3-13, (1927) [2] Baer, R, Linear algebra and projective geometry, (1952), Academic Press New York · Zbl 0049.38103 [3] Chacron, M, c-orderable division rings with involution, J. algebra, 75, 495-522, (1982) · Zbl 0482.16013 [4] \scM. Chacron, c-Orderable division rings with involution II, Canad. J. Math., in press. [5] Holland, S.S, Orderings and square roots in ∗-fields, J. algebra, 46, 207-219, (1977) · Zbl 0359.12023 [6] Holland, S.S; Holland, S.S, ∗-valuations and ordered ∗-fields, Trans. amer. math. soc., Erratum, 267, 333-243, (1981) · Zbl 0491.12026 [7] Krull, W, Allgemeine bewertungstheorie, J. reine angew. math., 167, 160-196, (1932) · JFM 58.0148.02 [8] Prestel, A, Quadratische semi-ordnungen und quadratische formen, Math. Z., 133, 319-342, (1973) · Zbl 0275.12013 [9] Rowen, L, Central simple algebras, Israel J. math., 29, 285-301, (1978) · Zbl 0392.16011 [10] Shiao, L.-S, Baer ordered ∗-rings, ()
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