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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

References

[AT] Albrecht, U. and Törner, G., Group rings and generalized valuations, Comm. Algebra 12(1984), 2243. 2272.Google Scholar
[A] André, J., Über Homomorphismen projektiver Ebenen, Abh. Math. Sem. Univ. Hamburg 34(1969), 98–114.Google Scholar
[BBT] Bessenrodt, C., Brungs, H.H. and Törner, G., Right Chain Rings, Schriftenreihe F.B. Math. 181, Duisburg, 1990.Google Scholar
[BT] Bessenrodt, C. and Tömer, G., Locally archimedean right-ordered groups and locally invariant valuation rings, J. Algebra 105(1987), 328–340.Google Scholar
[BG] Bmngs, H.H. and Gràter, J., Valuation rings infinite dimensional division algebras, J. Algebra 120(1989), 90–99.Google Scholar
[BT1] Brungs, H.H. and Tomer, G., Chain rings and prime ideals, Arch. Math. 27(1976), 253–260.Google Scholar
[BT2] Brungs, H.H., Extensions of chain rings, Math. Z. 185(1984), 93–104.Google Scholar
[CM] Cohn, P.M. and Mahdavi-Hezavehi, M., Extensions of valuations on skew fields, Proc. Ring Theory Week (ed. Van Oystaeyen, F.), Lecture Notes in Math. 825, Springer Verlag, Berlin, New York, 1980.Google Scholar
[D1] Dubrovin, N.I., Chain domains, Moscow Univ. Math. Bull. 36(1980), 56–60.Google Scholar
[D2] Dubrovin, N.I., An example of a chain prime ring with nilpotent elements, Math. USSR Sbornik 48(1984), 437— 444.Google Scholar
[D3] Dubrovin, N.I., preprint (Russian), 1992.Google Scholar
[G] Gräter, J., Über Bewertungen endlich dimensionaler Divisionsalgebren, Results Math. 7(1984), 54–57.Google Scholar
[HA] Hartmann, R, Stellen und Topologien von Schiefkörpern und Alternativkôrpern, Arch. Math. 51(1988), 274–282.Google Scholar
[H] Hausdorff, F., Mengenlehre, W. de Gruyter, Berlin, 1935.Google Scholar
[K] Kaplansky, I., Topological methods in valuation theory, Duke Math. J. 14(1947), 527–541.Google Scholar
[KO] Kokorin, A.I. and Kopytov, Y.M., Fully ordered groups, John Wiley and Sons, New York, 1974.Google Scholar
[KR] Krull, W., Allgemeine Bewertungstheorie, J. Reine Angew. Math. 167(1932), 160–196.Google Scholar
[L] Liepold, F., Zur Vervollständigung bewerteter Schiefkörper, Results Math. 19(1991), 122–142.Google Scholar
[M1] Mathiak, K., Bewertungen nichtkommutativer Körper, J. Algebra 48(1977), 217–235.Google Scholar
[M2] Mathiak, K., Valuation of skew fields and projective Hjelmslev spaces, Lecture Notes in Math. 1175, Springer Verlag, Berlin, 1986.Google Scholar
[O] Osofsky, B.L., Noncommutative rings whose cyclic modules have cyclic infective hulls, Pacific J. Math. 25(1968), 331–340.Google Scholar
[PA] Passman, D.S., The algebraic structure of group rings, John Wiley & Sons, New York, 1977.Google Scholar
[P] Posner, E.C., Left valuation rings and simple radical rings, Trans. Amer. Math. Soc. 107(1963), 458–465.Google Scholar
[R] Rado, F., Non-injective collineations on some sets in Desarguesian projective planes and extensions of noncommutative valuations, Aequationes Math. 4(1970), 307–321.Google Scholar
[RI] Ribenboim, P., Théorie des Valuations, Press de l'Université de Montréal, Montréal, 1968.Google Scholar
[S] Smirnov, D.M., Right-ordered groups, Algebra i Logika 5: 6(1966), 41–59.Google Scholar
[S1] Schröder, M., Bewertungsringe von Schiefkôrpern, Resultate und offene Problème, Results Math. 12(1987), 191–206.Google Scholar
[S2] Schröder, M., Über N.I. Dubrovin's Ansatz zur Konstruktion von nicht vollprimen Primidealen in Kettenringen,, Results Math. 17(1990), 296–306.Google Scholar
[SW] Shirvani, M. and Wehrfritz, B.A.F., Skew linear groups, Cambridge Univ. Press, New York, 1986.Google Scholar