  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 $\mathbb Q$, 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.
