Valuations of skew fields and projective Hjelmslev spaces
Let \(K\) be a skew field. By a valuation ring (sometimes called total subring) the author understands a subring \(V\) of \(K\) such that for all \(c\in K\) either \(c\in V\) or \(c^{-1}\in V\). This differs from the usual valuation ring on a skew field in not requiring \(a^{-1}Va=V\) for all \(a\in K^{\times}\). Just as the latter leads to Schilling valuations [\textit{O. F. G. Schilling}, Bull. Am. Math. Soc. 51, 297--304 (1945; Zbl 0061.05601)], so the former leads to a representation of \(K^{\times}\) by isotone mappings of a totally ordered set, briefly called ``valuations'' here [cf. the author, J. Algebra 48, 217--235 (1977; Zbl 0417.16006)]. The author studies these valuations (with numerous examples and exercises, including solutions), describing the ideal structure and topology defined by a valuation and an analogue of the approximation theorem for finite sets of valuations. A brief description of valued vector spaces is given and is applied to the study of projective Hjelmslev spaces induced by valued vector spaces. The join and meet in these spaces are replaced by ``union sets'' and ``intersection sets''; for these sets a dimension formula can be established (analogous to the usual case, but not holding for general Hjelmslev spaces over local rings). An appendix by J. Graeter deals with locally invariant valuations, i.e. valuations such that \(xP(x)=P(x)x\) for all \(x\) in the maximal ideal of the valuation ring \(V\), where \(P(x)\) denotes the minimal completely prime ideal of \(V\) containing \(x\). For these valuations Graeter is able to prove an approximation theorem under weaker hypotheses, as in \textit{P. Ribenboim}, Théorie des valuations [Presses de l'Université de Montréal (1965; Zbl 0139.26201)].
scientific article; zbMATH DE number 3937288