Real Algebraic and Analytic Geometry |

$n$-real valuations and the higher level version of the Krull-Baer theorem.

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Submission: 2004, September 23.

*Abstract:
Cimpri\v c gave examples of division rings containing an
ordering of level $2m$ but not of level $m$ for $m \in \N$. His examples
were quite complicated. We give substantially simplified examples in Section
2.
In Sections 3 and 4 we investigate this phenomenon using valuation theory.
We define almost real and $n$??real valuations and study liftings of orderings
from the residue division ring to the original division ring. Such liftings
are not always possible (as is the case in the commutative setting), but
we give a necessary and sufficient condition for a lifting to exist. We
also prove a suitable generalization of the Baer??Krull theorem.
Finally, in the last section we use our examples and the theory developed
to answer a question given by Marshall \& Zhang.
.*

Mathematics Subject Classification (2000): 14P99, 06Fxx.

Keywords and Phrases: orderings of higher level, division rings, valuations, signatures, real places.

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