An environment of quasi-valuation domains (Q1076730)

The author defines a generalization of valuation domains, called quasi- valuation domains. As value group he uses an ordered Abelian group which carries a second (pre)order related to the first one and such that both order relations are compatible with the group operations. If V is a valuation domain of a field K, with maximal ideal M, then for any subdomain W of V with quotient field K there are a number of equivalent conditions to W being a quasi-valuation domain; we quote only two of them: W (which of course is local) is a quasi-valuation domain if and only if its maximal ideal is equal to M; or equivalently, if for any prime ideal P of W the maximal ideal of \(V_ P\) is equal to P. - The author derives a number of technical details about quasi-valuation domains as well as about pre-valuation domains, which are defined by the condition that V is integral over W. Among the more substantial theorems there is one showing that any quasi-valuation domain is seminormal in the sense of \textit{C. Traverso} [Ann. Sc. Norm. Super., Pisa, Sci. Fis. Mat., III. Ser. 24(1970), 585-595 (1971; Zbl 0205.505)].