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A Short Note on the Higher Level Version of the Krull--Baer Theorem

  Published:2010-08-19
 Printed: Jun 2011
  • Dejan Velušček,
    University of Ljubljana, Faculty of Mathematics and Physics, Department of Mathematics, Ljubljana, Slovenia
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Abstract

Klep and Velu\v{s}\v{c}ek 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 Velu\v{s}\v{c}ek, and thus complete the generalization of the Krull--Baer theorem for preorderings.
Keywords: orderings of higher level, division rings, valuations orderings of higher level, division rings, valuations
MSC Classifications: 14P99, 06Fxx show english descriptions None of the above, but in this section
Ordered structures
14P99 - None of the above, but in this section
06Fxx - Ordered structures
 

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