Valuation methods in division rings
Let \(R\) be a \(Z\)-filtered ring whose associated graded ring is an Ore domain. Then \(R\) can be embedded in a skew field. This was proved by the reviewer [Proc. Lond. Math. Soc., III. Ser. 11, 511-530 (1961; Zbl 0104.032)]. The author here gives another relatively short proof of this result; this brings the number of proofs in print to five [see \textit{P. M. Cohn}, Skew Fields (Encycl. Math. Appl. 57, Cambridge Univ. Press 1995)]. The author's proof proceeds by adjoining a central indeterminate \(t\), extending the given filtration \(v\) by putting \(v (\sum t^i a_i) = \min \{v (a_i) + i\}\) and then showing that any ring \(A\) with a central element \(t\) such that \(A/(t)\) is an Ore domain and \(\bigcap (t^n) = 0\), can be embedded in a skew field \(D\); the valuation \(v\) extends to \(D\) and \(AA^{-1}\) is dense in \(D\) while the skew field of fractions of \(A/(t)\) appears as the residue-class field. In further applications the valuation \(v\) is assumed quasi-abelian, i.e. the associated graded ring is commutative. For a skew field \(D\) with a discrete quasiabelian valuation trivial on the prime subfield the author shows (i) if \(\text{char }D = 0\), then the centre \(Z\) of \(D\) is relatively algebraically closed in \(D\); (ii) if the residue-class field has finite characteristic \(p\), then every commutative finite-dimensional extension field of \(Z\) has prime power dimension.
scientific article; zbMATH DE number 827380