We define a kind of non-commutative Krull domains using serial over-rings, which turn out to be localizations at prime ideals of height one. We globalize some properties of valuation rings to these so-called S-Krull domains in particular, every element is normalizing. We study divisorial ideals and introduce the class group and establish that the class group (as well as the notion of S-Krull domain) behaves well under localization at T-functors. We point out some special cases and the relation to existing theories of R. Fossum, M. Chamarie and H. Marubayashi.