Completed group algebras without zero divisors
Let p be a prime, and let G be a torsion free analytic pro-p group; that is, G is a torsion free pro-p group admitting the structure of an analytic manifold over \(Q_ p\) with analytic group operations. By a result of \textit{A. Lubotzky} and \textit{A. Mann} [J. Algebra 105, 484-505, 506-515 (1987; Zbl 0626.20010, Zbl 0626.20022)], a pro-p group is analytic if and only if there is a finite bound on the cardinality of a generating set of its open subgroups. (Such groups include the poly-p- adic groups, those pro-p groups having a finite subnormal series of closed subgroups \(1=G_ 0\subset...G_ i\subset...\subset G_ n=G\) with \(G_{i+1}/G_ i\) isomorphic either to the p-adic integers \({\mathbb{Z}}_ p\) or to the cyclic group of order p, for all p.) Let \({\mathbb{Z}}_ pG\) denote the completion \(\lim_{\leftarrow}{\mathbb{Z}}_ p[G/N]\), where G/N ranges over the finite images of G. The main result of this note is that \({\mathbb{Z}}_ pG\) is a domain. Since \({\mathbb{Z}}_ p[G]\) embeds in \({\mathbb{Z}}_ pG\), the same conclusion applies to the ordinary group ring too. Being a projective limit of local rings, \(Z_ pG\) is a local ring. Appealing largely to results of \textit{M. Lazard} [Publ. Math., Inst. Hautes Etud. Sci. 26, 389-603 (1965; Zbl 0139.023)], the author shows that \({\mathbb{Z}}_ pG\) is Noetherian of finite global dimension. The desired conclusion now follows from a result of \textit{R. Walker} [Proc. Lond. Math. Soc., III. Ser. 24, 27-45 (1972; Zbl 0224.16004)].
finite subnormal series of closed subgroups
scientific article; zbMATH DE number 4095672