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 2m 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 [

Higher Product Pythagoras Numbers of Skew Fields

“…In order to either push orderings from D to k v or to pull orderings from k v back to D, we introduce n-real valuations (see [7] for more details).…”

Central Extensions of n-Ordered Division Rings

“…Now let us recall a result from Klep and Velušček (2004) used frequently in the sequel. In Klep and Velušček (2004) we studied the following problem: given a Downloaded by [University of Birmingham] at 16:26 18 November 2014…”

“…v is called n-real if there exists an ordering P of k v such that × v ∩ 2n D × ⊆ P. We also say that P (or the corresponding signature) is n-compatible with v. The following Krull-Baer type result is given in Klep and Velušček (2004): v is called n-real if there exists an ordering P of k v such that × v ∩ 2n D × ⊆ P. We also say that P (or the corresponding signature) is n-compatible with v. The following Krull-Baer type result is given in Klep and Velušček (2004):…”

“…Theorem (Theorem 9 in Klep and Velušček, 2004). Let v be a valuation of D and let be a signature of level n of k v which is n-compatible with v. Then:…”

A Variant of Higher Product Levels of Integral Domains and *-Domains

“…Therefore, for a skew-field D and number n, we have besides s n (D) at least two other related invariants, ms n (D) and ps n (D), which are defined as the number of terms in the shortest representation of −1 as a sum of products (in the case of ms n ) or permuted products (in the case of ps n ) of n-th powers of elements from D. The invariant ps n (D) for n > 2 was introduced in [4] and studied later in [10] and [6]. The motivation for studying ps n comes from the theory of higher level orderings.…”