Applications of generalized universal valuations

We introduce a generalization of the universal valuation semiring defined by Jeffrey and Noah Giansiracusa. We then explicitly characterize the additive structure of this semiring and show that, when applied to Q, this characterization gives the non-Archimedean case of Ostrowski's theorem. We conclude with examples of non-commutative valuations and their applications, such as the detection of the existence of representations of rings in ultrametric vector spaces.