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Prime length of crossed products. (English) Zbl 0673.16005

Let R*G be a crossed product with G a polycyclic-by-finite group and R a right Noetherian ring. This paper gives an upper bound for the prime length \((=\) classical Krull dimension), dim(R*G), of R*G in terms of the prime length of R and a group theoretical invariant of G, the so-called plinth length, p(G), of G. Specifically, the author shows that \(\dim (R*G)<(\dim (R)+1)\cdot (p(G)+1)\). This extends a result of J. E. Roseblade [Proc. Lond. Math. Soc., III. Ser. 36, 385-447 (1978; Zbl 0391.16008)] for polycyclic group algebras and improves a certain bound for polycyclic crossed products given by D. S. Passman [Trans. Am. Math. Soc. 301, 737-759 (1987; Zbl 0619.16007)]. The methods used in the proof come from Passman’s and Roseblade’s work.
Reviewer: M.Lorenz

MSC:

16S34 Group rings
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
20F19 Generalizations of solvable and nilpotent groups
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
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References:

[1] Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. · Zbl 0368.16003
[2] Donald S. Passman, Group rings of polycyclic groups, Group theory, Academic Press, London, 1984, pp. 207 – 256. · Zbl 0555.16007
[3] D. S. Passman, Prime ideals in polycyclic crossed products, Trans. Amer. Math. Soc. 301 (1987), no. 2, 737 – 759. · Zbl 0619.16007
[4] J. E. Roseblade, Prime ideals in group rings of polycyclic groups, Proc. London Math. Soc. (3) 36 (1978), no. 3, 385 – 447. , https://doi.org/10.1112/plms/s3-36.3.385 J. E. Roseblade, Corrigenda: ”Prime ideals in group rings of polycyclic groups” (Proc. London Math. Soc. (3) 36 (1978), no. 3, 385 – 447), Proc. London Math. Soc. (3) 38 (1979), no. 2, 216 – 218. · Zbl 0406.16008 · doi:10.1112/plms/s3-38.2.216-s
[5] B. A. F. Wehrfritz, Infinite linear groups. An account of the group-theoretic properties of infinite groups of matrices, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Matematik und ihrer Grenzgebiete, Band 76. · Zbl 0261.20038
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