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Primitive group rings and Noetherian rings of quotients. (English) Zbl 0562.16005
Let k be a field and G a nontrivial countable hypercentral group with centre Z. Let H be a torsionfree subgroup of G such that some positive power of every element of G lies in H and every finite rank subgroup of H is finitely generated, and let \(Z_ 0=Z\cap H\). - It is proved that the following statements are equivalent: (a) the group ring kG is primitive; (b) k is countable, G is torsionfree and there exists an abelian subgroup A of G, of infinite rank, with \(A\cap Z=1\); (c) k is countable, G is torsionfree and the partial quotient ring \((kH)(kZ_ 0)^{-1}\) is not noetherian.
The implication (b)\(\Rightarrow (a)\) is a consequence of one of Zalesskij’s intersection theorems [see K. A. Brown, Arch. Math. 36, 404-413 (1981; Zbl 0464.16009), Lemma 3.1] and the implication (a)\(\Rightarrow (c)\) is relatively straight forward, proceeding in two steps, firstly showing that kH is primitive and secondly using a strong form of P. Hall’s generic flatness [see P. Hall, Proc. Lond. Math. Soc., III. Ser. 9, 595-622 (1959; Zbl 0091.02501), Lemma 4,1, and K. A. Brown, J. Lond. Math. Soc., II. Ser. 26, 425-434 (1982; Zbl 0511.16011)]. The deepest part of the theorem is the fact that (c) implies (b). The strategy employed for proving certain rings of fractions are noetherian is embodied in the following result: let \[ S_ 1\subseteq S_ 2\subseteq...\subseteq S_{\alpha}\subseteq S_{\alpha +1}\subseteq...\subseteq \cup_{\lambda <\rho}S_{\lambda}=R \] be an ascending chain of right noetherian subrings of a ring R such that R is a flat left \(S_{\lambda}\)-module for all \(\lambda\) ; then R is right noetherian if and only if given a nonzero right R-module M there exists an ordinal \(\lambda <\rho\) and a nonzero \(S_{\lambda}\)-submodule N of M with \(NR=N\otimes_{S_{\lambda}}R\).
Reviewer: P.F.Smith

16S34 Group rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20E07 Subgroup theorems; subgroup growth
20F19 Generalizations of solvable and nilpotent groups
16P50 Localization and associative Noetherian rings
16P40 Noetherian rings and modules (associative rings and algebras)
Full Text: DOI
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