Geodesic
A Short Note on the Higher Level Version of the Krull–Baer Theorem
Canadian mathematical bulletin, Tome 54 (2011) no. 2, pp. 381-384

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

Klep and Velušček generalized the Krull–Baer theorem for higher level preorderings to the non-commutative setting. A $n$ -real valuation $v$ on a skew field $D$ induces a group homomorphism $\overline{v}$ . A section of $\overline{v}$ is a crucial ingredient of the construction of a complete preordering on the base field $D$ such that its projection on the residue skew field ${{k}_{v}}$ equals the given level 1 ordering on ${{k}_{v}}$ . In the article we give a proof of the existence of the section of $\overline{v}$ , which was left as an open problem by Klep and Velušček, and thus complete the generalization of the Krull–Baer theorem for preorderings.
DOI : 10.4153/CMB-2010-095-0
Mots-clés : 14P99, 06Fxx, orderings of higher level, division rings, valuations 
Velušček, Dejan. A Short Note on the Higher Level Version of the Krull–Baer Theorem. Canadian mathematical bulletin, Tome 54 (2011) no. 2, pp. 381-384. doi: 10.4153/CMB-2010-095-0
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[1] [1] Becker, E., Summen n-ter Potenzen in Körpern. J. Reine Angew. Math. 307/308(1979), 8–30. Google Scholar

[2] [2] Klep, I., Velušček, D., n-real valuations and the higher level version of the Krull–Baer theorem. J. Algebra 279(2004), no. 1, 345–361. doi:10.1016/j.jalgebra.2004.05.012 Google Scholar

[3] [3] Fuchs, L., Infinite abelian groups. Vol. I. Pure and Applied Mathematics, 36, Academic Press, New York-London, 1970. Google Scholar

[4] [4] Schilling, O. F. G., The theory of valuations, Mathematical Surveys, 4, American Mathematical Society, New York, 1950. Google Scholar

[5] [5] Powers, V., Holomorphy rings and higher level orders on skew fields. J. Algebra 136(1991), no. 1, 51–59. doi:10.1016/0021-8693(91)90063-E Google Scholar

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