Valuations and valuation spectra for division rings and central simple algebrasVukšić, Lara (Avtor) Klep, Igor (Mentor) Weyl algebranoncommutative valuationsskew polynomial ringsorderingsextensions of valuationsextensions of orderings.We first introduce the notion of strongly abelian valuations on noncommutative division rings. The value group and residue field of a strongly abelian valuation are both commutative. Then we classify all valuations on the real Weyl algebra with real residue field. These valuations are all strongly abelian. Then we classify all valuations with real residue field on a ring extension of the real Weyl algebra with the real closure of the field of rational functions, where in one case, we use compactness theorem from model theory. These valuations are also strongly abelian. Using this classification, we describe all valuations on the real Weyl algebra that extend to the above mentioned ring extension. We show that Kaplansky's theorem which states that all extensions by limits of pseudo-convergent sequences are immediate does not hold for noncommutative division rings in general. We describe all orderings on the real Weyl algebra and its extension with the real closure of the field of rational functions. Lastly, we describe the possible application of the construction of valuations on the real Weyl algebra to other skew polynomial rings.20232023-09-30 08:15:02Doktorsko delo/naloga151176UDK: 512VisID: 138261COBISS_ID: 166658819sl