Valuation methods in group rings and in skew fields. I
The author is the expert on applying valuation theoretic methods to group rings, enveloping algebras of Lie algebras, and skew fields. In this paper, he constructs a family of discrete valuations in group rings of residually torsion-free nilpotent groups and extends these valuations to the Malcev-Neumann power series skew field of these group rings. The main results here are quite technical in nature, so we mention just two of the many theorems in this long paper. Theorem~I. Let \(H\) be a group with an N-series \(\{H_i\}\) and let \(K\) be a (commutative) field of characteristic \(p\). Assume that \(\bigcap_{i=1}^\infty H_i=1\) and that all factors \(H_i/H_{i+1}\) are torsion-free Abelian with no elements of infinite \(p\)-height. Then, using the filtration defined by the series, we have \(\text{gr}(KH)\cong K\otimes\text{gr}(ZH)\). This, in turn, yields results on the enveloping algebras of certain Lie algebras associated to \(H\). Theorem~III. Let \(R\) be a ring and let \((t)\) be the ideal generated by the central regular element \(t\), with \(\bigcap_{n=1}^\infty(t)^n=0\). If \(\overline R=R/(t)\) is a semiprime Goldie ring, then the \(t\)-completion \(\widetilde R\) of \(R\) can be embedded in a semiprime Artinian ring. Furthermore, if \(\overline R\) is an Ore domain with skew field of fractions \({\mathcal D}(\overline R)\), then \(R\) can be embedded in a skew field \({\mathcal D}(R)\) having a \(t\)-adic valuation with residue skew field \({\mathcal D}(\overline R)\). This is a key ingredient in the proofs of several of the main group ring results.
residually torsion-free nilpotent groups
scientific article; zbMATH DE number 1868058
scientific article