Valuations in groups and rings

Orderings and Valuations on Twisted Polynomial Rings

The real spectra of certain \twisted" polynomial algebras A over R are examined. In certain cases, including Weyl algebras and universal enveloping algebras of solvable Lie algebras, the stability indices of the residue spaces of Sper(A) are estimated. The results show that the Brr ocker-Scheiderer theory of minimal generation of constructible sets applies to Sper(A) in these cases. An attempt ...

Extending Real Valuations to Skew Polynomial Rings

Let D be a division ring, T be a variable over D, σ be an endomorphism of D, δ be a σ-derivation on D and R = D[T ;σ, δ] the left skew polynomial ring over D. We show that the set (V alν(R), ) of σ-compatible real valuations which extend to R a fixed proper real valuation ν on D has a natural structure of parameterized complete non-metric tree, where is the partial order given by μ μ̃ if and onl...

Approximation theorems for valuations on commutative rings

Valuations and Higher Level Orders in Commutative Rings

Valuation theory is one of the main tools for studying higher level orders and the reduced theory of forms over fields, see, for example [BR]. In [MW], the theory of higher level orders and reduced forms was generalized to rings with many units and many of the results for fields carried over to this setting. While it seems desirable to extend these results further, the techniques used for rings...

Realization of a Certain Class of Semi-Groups as Value Semi-Groups of Valuations