Zur Vervollständigung bewerteter Schiefkörper. (On the completion of valued skew fields) (Q755795)

A valuation ring B in a skew field K is a subring B with \(K=B\cup (B\setminus \{0\})^{-1}\). Then K is a topological skew field with respect to the ``place topology'' \(\tau_ B\), which is generated by the sets aBa with \(a\in K^*\), cf. \textit{P. Hartmann} [Arch. Math. 51, 274- 282 (1988; Zbl 0673.16024)] and \textit{M. Schröder} [Result. Math. 12, 191-206 (1987; Zbl 0639.16011)]. If \(\tau_ B\) is a v-topology, then the ring completion \(\hat K\) of \((K,\tau_ B)\) is again a skew field. The author proves the following quite surprising result: if \(\tau_ B\) is not a v-topology and if the linearly ordered sets \(\{\) aB\(| a\in K^*\}\) and \(\{\) Ba\(| a\in K^*\}\) both have coinitiality type \(\omega_ 0\), then \(\hat K\) is not a skew field and contains precisely two maximal ideals. Examples show that the countability of the coinitiality type is an essential assumption.