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Holomorphy rings and higher level orders on skew fields. (English) Zbl 0715.12002

The author extends Becker’s theory of higher level orders and preorders to division rings as far as this theory can go. Her main tool is an extension of valuation theory from fields to division rings. She also finds a sensible generalization of the concept of Prüfer ring to subrings of division rings. Finally, she shows that all her commutative theorems [Math. Z. 198, No. 4, 545–554 (1988; Zbl 0627.10014)] are valid for division rings.

MSC:

12E15 Skew fields, division rings
16W80 Topological and ordered rings and modules
11E81 Algebraic theory of quadratic forms; Witt groups and rings
12J15 Ordered fields
16K20 Finite-dimensional division rings
06F25 Ordered rings, algebras, modules

Citations:

Zbl 0627.10014
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Full Text: DOI

References:

[1] Becker, E., Partial orders on a field and valuation rings, Comm. Algebra, 7, 1933-1976 (1979) · Zbl 0432.12011
[2] Becker, E., Valuations and real places in the theory of formaly real fields, (Géométrie Algébrique Réelle et Formes Quadratiques. Géométrie Algébrique Réelle et Formes Quadratiques, Lecture Notes in Mathematics, No. 959 (1982), Springer-Verlag: Springer-Verlag Berlin/New York)
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[12] Powers, V., Higher level reduced Witt rings of skew fields, Math. Z., 198, 545-554 (1988) · Zbl 0627.10014
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