Ideals in group rings of soluble groups of finite rank.

*(English)*Zbl 0532.16007The 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 \(\Delta(G)=1\).

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 |

##### Keywords:

primitive group rings; gamma-ideals of group algebras; standardisers of ideals; minimal primes; faithful almost standardised primes; orbital elements; maximum condition on ideals; plinths; valuation sphere; soluble groups of finite rank; annihilator-free prime ideals
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\textit{C. J. B. Brookes}, Math. Proc. Camb. Philos. Soc. 97, 27--49 (1985; Zbl 0532.16007)

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##### References:

[1] | DOI: 10.1112/plms/s3-41.3.439 · Zbl 0448.20029 · doi:10.1112/plms/s3-41.3.439 |

[2] | Bieri, Soluble Groups with Coherent Group Rings pp 235– (1979) · Zbl 0425.20028 |

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