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Ideals in group rings of soluble groups of finite rank. (English) Zbl 0532.16007
The bulk of this paper is a study of $$\Gamma$$-ideals of the group algebra $$kA$$ where $$\Gamma$$ is a group acting on a torsion-free abelian group $$A$$ of finite rank. Define the standardiser of an ideal $$I$$ of $$kA$$ to be the subgroup of $$\Gamma$$ of elements $$\gamma$$ such that $$I\cap kA_ 1=I^{\gamma}\cap kA_ 1$$ for some finitely generated (f.g.) subgroup $$A_ 1$$ (dependent on $$\gamma$$) of rank $$r(A)$$. Every $$\Gamma$$-ideal has at least one minimal prime over it almost standardised by $$\Gamma$$. This is significant because faithful almost standardised primes of $$kA$$ are controlled by $$\Delta_{\Gamma}(A)$$, the subgroup of $$\Gamma$$-orbital elements of $$A$$ (Theorem A). The proof is based upon that of Theorem D of J. E. Roseblade [Proc. Lond. Math. Soc. (3) 36, 385–447 (1978; Zbl 0391.16008)], the analogous result for f.g. $$A$$. This was a generalisation of a theorem of G. M. Bergman [Trans. Am. Math. Soc. 157, 459–470 (1971; Zbl 0197.17102)] which itself may be extended to the finite rank case. $$A$$ is said to be a plinth if it is rationally irreducible as a $${\mathbb{Z}}\Gamma_ 1$$-module for all $$\Gamma_ 1$$ of finite index in $$\Gamma$$. Part of Theorem C states: Let $$A$$ be a plinth which has the maximum condition on $$\Gamma$$-invariant subgroups. Then $$kA$$ has the maximum condition on $$\Gamma$$-ideals and a non-zero $$\Gamma$$-ideal $$I$$ satisfies $$\dim_ k(kA/I)<\infty$$. Furthermore if $$\Gamma$$ is f.g. nilpotent and $$G$$ is any group containing $$A$$ as a normal subgroup with $$G/A\cong\Gamma$$ then $$kG$$ has the maximum condition on ideals. This is significant because it provides numerous examples on non-Noetherian rings with the maximum condition on ideals. Moreover there are groups $$G$$ with $$kG$$, but not $${\mathbb{Z}}G$$, having that property.
On the other hand the results about $$\Gamma$$-ideals do not carry over from the f.g. to the finite rank case. For fields of characteristic zero faithful primes need not be controlled by $$\Delta_{\Gamma}(A)$$ even when $$A$$ is the direct product of two plinths. For f.g. abelian $$\Gamma$$ a condition is obtained in terms of the valuation sphere introduced by R. Bieri and R. Strebel [Proc. Lond. Math. Soc. (3) 41, 439–464 (1980; Zbl 0448.20029)], sufficient for the existence of this departure from polycyclic behaviour. This condition also proves to be a necessary one when $$\Gamma$$ is infinite cyclic. This effectively provides a characterisation of the actions of an infinite cyclic group $$\Gamma$$ for which all faithful $$\Gamma$$-primes are annihilator-free.
Two applications of this study of $$\Gamma$$-ideals are:
Corollary E. Let $$G$$ be a soluble group of finite rank with no non-trivial finite normal subgroups and $$k$$ be a field. Then all annihilator-free prime ideals of $$kG$$ are controlled by $$\Delta(G)$$.
Theorem F. Let $$G$$ be a soluble group of finite rank and $$k$$ be a non-absolute field. Then $$kG$$ is primitive if $$\Delta(G)=1$$.
Reviewer: C. J. B. Brookes

MSC:
 16S34 Group rings 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 16D25 Ideals in associative algebras 20F16 Solvable groups, supersolvable groups 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
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References:
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