N-real valuations and the higher level version of the Krull–Baer theorem

Abstract

In [Comm. Algebra 29 (2001) 193–201] Cimprič gave examples of division rings containing an ordering of level 2 m but not of level m for m∈ 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 and Zhang [J. Algebra 212 (1) (1999) 190–207].