Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T06:29:13.902Z Has data issue: false hasContentIssue false
  • English
  • Français

Value Groups and Distributivity

Published online by Cambridge University Press:  20 November 2018

H. H. Brungs  and
J. Gräter
H. H. Brungs
Affiliation:
Department of Mathematics, University of Alberta, Edmonton, AlbertaT6G 2G1
J. Gräter
Affiliation:
Institut für Analysis, Abt. für Topologie u. Grundlagen der Analysis, Technische Universität, Pockelsstrasse 14, D-3300 Braunschweig, Germany
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let F be a skew field with a valuation (also called total) subring B, i.e. x in F\ B implies x-1 in B. Such rings are useful not only in the investigation and construction of division algebras (see for example [5],[6],[12]) but also in geometry ([15]).

Associated with B is an invariant subring R of F and a value group G. We investigate the relationship between properties like the distributivity of R and properties like being lattice ordered of G.

Keywords

16A1416A10061512F10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 6 , 01 December 1991 , pp. 1150 - 1160
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Bessenrodt, Chr. and Törner, G., Locally archimedean right-ordered groups and locally invariant valuation rings, J. Algebra 105(1987), 328–340.Google Scholar
2. Birkhoff, G., Lattice theory. AMS Colloquium Publications, 25, Providence, R.I., 1967.Google Scholar
3. Brungs, H.H., Bezout domains and rings with a distributive lattice of right ideals, Can. J. Math. 38(1986), 286–303.Google Scholar
4. Brungs, H.H. and Törner, G., Extensions of chain rings, Math. Z. 185(1984), 93–104.Google Scholar
5. Brungs, H.H. and Gräter, J., Valuation rings in finite-dimensional division algebras, J. Algebra 120(1989), 90–99.Google Scholar
6. Cohn, P.M. and Mahdavi-Hezavehi, M., Extensions of valuations on skew fields, in Ring Theory Antwerp 1980, ed. F. Van Oystaeyen, L.N.M. 825, Springer-Verlag, Berlin, 1980,28–41.Google Scholar
7. Endler, O., Valuation theory. Springer-Verlag, Berlin-New York, 1972.Google Scholar
8. Gräter, J., Zur Théorie nicht kommutativer Prüferringe, Arch. Math. 41(1983), 30–36.Google Scholar
9. Gräter, J., Lokalinvariante Bewertungen, Math. Z. 192(1986), 183–194.Google Scholar
10. Gräter, J., On noncommutative Prüfer rings, Arch. Math. 46(1986), 402-407.Google Scholar
11. Gräter, J. , Valuations on finite-dimensional division algebras and their value groups, Arch. Math. 51(1988), 128–140.Google Scholar
12. Jacob, B. and Wadsworth, A.R., A new construction of noncrossedproduct algebras, Trans. Amer. Math. Soc. 293(1986), 693–721.Google Scholar
13. Mathiak, K., Bewertungen nicht kommutativer Körper, J. Algebra 48(1977), 217–235.Google Scholar
14. Mathiak, K.,Zur Bewertungstheorie nicht kommutativer Körper, J. Algebra 73(1981), 586–600.Google Scholar
15. Mathiak, K., Valuations of skew fields and projective Hjelmslev spaces. L.N.M. 1175, Springer-Verlag, Berlin- New York, 1986.Google Scholar
16. Neumann, B.H., On ordered division rings, Trans. Amer. Math. Soc. 66(1949), 202–252.Google Scholar
17. Ribenboim, P., Algebraic numbers. John Wiley and Sons, 1972.Google Scholar
18. Schilling, O.F., Noncommutative valuations, Bull. Amer. Math. Soc. 51(1945), 297–304.Google Scholar
19. Schilling, O.F., The theory of valuations. Math. Surveys, No. 4, Amer. Math. Soc. Providence, R.I., 1950.Google Scholar