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Prime Segments of Skew Fields

Published online by Cambridge University Press:  20 November 2018

H. H. Brungs  and
M. Schröder
H. H. Brungs
Affiliation:
Department of Mathematics University of Alberta Edmonton, Alberta
M. Schröder
Affiliation:
Fachbereich Mathematik Universität Duisburg Lotharstr.65 D-47048 Duisburg Germany
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Abstract

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An additive subgroup P of a skew field F is called a prime of F if P does not contain the identity, but if the product xy of two elements x and y in F is contained in P, then x or y is in P. A prime segment of F is given by two neighbouring primes P1 ⊃ P2; such a segment is invariant, simple, or exceptional depending on whether A(P1) = {a ∈ P1 | P1aP1 ⊂ P1} equals P1, P2 or lies properly between P1 and P2. The set T(F) of all primes of F together with the containment relation is a tree if |T(F)| is finite, and 1 < |T(F)| < ∞ is possible if F is not commutative. In this paper we construct skew fields with prescribed types of sequences of prime segments as skew fields F of fractions of group rings of certain right ordered groups. In particular, groups G of affine transformations on ordered vector spaces V are considered, and the relationship between properties of Dedekind cuts of V, certain right orders on G, and chains of prime segments of F is investigated. A general result in Section 4 describing the possible orders on vector spaces over ordered fields may be of independent interest.

Keywords

16W6016S3516L3016W8020F60
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 6 , 01 December 1995 , pp. 1148 - 1176
Copyright
Copyright © Canadian Mathematical Society 1995

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