Roseblade, J. E. Group rings of polycyclic groups. (English) Zbl 0285.20008 J. Pure Appl. Algebra 3, 307-328 (1973). Phillip Hall in 1959 proved that if \(G\) is a finitely generated nilpotent by finite group and \(V\) is an irreducible \(F[G]\) module where \(F\) is algebraic over its prime subfield and of characteristic \(p>0\), then \(V\) is finite dimensional over \(F\) [Proc. Lond. Math. Soc. (3) 9, 592–622 (1959; Zbl 0091.02501)]. In that paper, Hall asked if the same is true if \(G\) is polycyclic by finite. In this very important paper, Hall’s question is answered affirmatively. The proof involves a careful study of the structure of \(V\) as an \(F[A]\) module where \(A\) is an infinite abelian normal subgroup. This is a generalization of Hall’s idea of studying the structure as an \(F[x,x^{-1}]\) module where \(x\) is central in \(G\). The proof relies on a theorem of G. M. Bergman [Trans. Am. Math. Soc. 157, 459–469 (1971; Zbl 0197.17102)] at a critical point. Hall used his theorem to show that finitely generated abelian by nilpotent by finite groups are residually finite. The analogous theorem for finitely generated abelian by polycyclic by finite groups has been recently proved by A. V. Jategaonkar [J. Pure Appl. Algebra 4, 337–343 (1974; Zbl 0297.20013)]. Reviewer: Robert L. Snider (Blacksburg) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 18 ReviewsCited in 77 Documents MSC: 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 16S34 Group rings 20E15 Chains and lattices of subgroups, subnormal subgroups Citations:Zbl 0091.02501; Zbl 0197.17102; Zbl 0297.20013 PDFBibTeX XMLCite \textit{J. E. Roseblade}, J. Pure Appl. Algebra 3, 307--328 (1973; Zbl 0285.20008) Full Text: DOI References: [1] Atiyah, M. F.; Macdonald, I. G., Introduction to Commutative Algebra (1969), Addison-Wesley: Addison-Wesley Reading, Mass · Zbl 0175.03601 [2] Bergman, G. M., The logarithmic limit-set of an algebraic variety, Trans. Am. Math. Soc., 157, 459-469 (1971) · Zbl 0197.17102 [3] Gaschutz, W., Über die qf-Untergruppe endlicher Gruppen, Math. Z., 58, 160-170 (1953) · Zbl 0050.02202 [4] Hall, P., Finiteness conditions for soluble groups, Proc. London Math. Soc., 4, 3, 419-436 (1954) · Zbl 0056.25603 [5] Hall, P., On the finiteness of certain soluble groups, Proc. London Math. Soc., 9, 3, 595-622 (1959) · Zbl 0091.02501 [6] Hall, P., The Frattini subgroups of finitely generated groups, Proc. London Math. Soc., 11, 3, 327-352 (1961) · Zbl 0104.02201 [7] Hall, P., On non-strictly simple groups, Proc. Cambridge Phil. Soc., 59, 531-553 (1963) · Zbl 0118.03601 [8] Hirsch, K. A., On infinite soluble groups III, Proc. London Math. Soc., 49, 2, 184-194 (1946) · Zbl 0063.02021 [9] Hirsch, K. A., On infinite soluble groups V, J. London Math. Soc., 29, 250-251 (1954) · Zbl 0055.01603 [10] Krull, W., Jacobsonsches Radikal und Nullstellensatz, Proc. Intern. Congr. of Mathematicians, 2, 56-64 (1950) [11] Levič, E. M., Math. Dokl., 10, 1299-1301 (1969) [12] Levitzki, J., On multiplicative systems, Compositio Math., 8, 76-80 (1950) · Zbl 0033.34801 [13] Mal’cev, A. I., Am. Math. Soc. Transl., 2, 2, 1-21 (1956) [14] Neumann, B. H., Groups covered by permutable subsets, J. London Math. Soc., 29, 236-248 (1954) · Zbl 0055.01604 [15] Roseblade, J. E., The integral group rings of hypercentral groups, Bull. London Math. Soc., 3, 351-355 (1971) · Zbl 0243.20005 [16] Swan, R. G., \(K\)-Theory of Finite Groups and Orders, Lecture Notes in Math., 149 (1970), Springer: Springer Berlin · Zbl 0205.32105 [17] Zaleskiǐ, A. L., On the structure of finitely generated modules over the group algebra of a polycyclic group, Dokl. Akad. Nauk BSSR, 14, 977-980 (1970) · Zbl 0222.20009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.