Geodesic
Prime Segments of Skew Fields
Canadian journal of mathematics, Tome 47 (1995) no. 6, pp. 1148-1176

Voir la notice de l'article provenant de la source Cambridge University Press

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 P 1 ⊃ P 2; such a segment is invariant, simple, or exceptional depending on whether A(P 1) = {a ∈ P 1 | P 1 aP 1 ⊂ P 1} equalsP 1, P 2 or lies properly between P 1 and P 2. 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.
DOI : 10.4153/CJM-1995-059-3
Mots-clés : 16W60, 16S35, 16L30, 16W80, 20F60 
Brungs, H. H.; Schröder, M. Prime Segments of Skew Fields. Canadian journal of mathematics, Tome 47 (1995) no. 6, pp. 1148-1176. doi: 10.4153/CJM-1995-059-3
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