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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Analogs of Clifford's theorem for polycyclic-by-finite groups


Author: Martin Lorenz
Journal: Trans. Amer. Math. Soc. 254 (1979), 295-317
MSC: Primary 20C07
DOI: https://doi.org/10.1090/S0002-9947-1979-0539920-X
MathSciNet review: 539920
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Abstract: Let P be a primitive ideal in the group algebra $ K[G]$ of the polycyclic group G and let N be a normal subgroup of G. We show that among the irreducible right $ K[G]$-modules with annihilator P there exists at least one, V, such that the restricted $ K[N]$-module $ {V_N}$ is completely reducible, a sum of G-conjugate simple $ K[N]$-submodules. Various stronger versions of this result are obtained. We also consider the action of G on the factor $ K[N]/P \cap K[N]$ and show that, in case K is uncountable, any ideal I of $ K[N]$ satisfying $ { \cap _{g \in G}}{I^g}\, = \,P\, \cap \,K[N]$ is contained in a primitive ideal Q of $ K[N]$ with $ { \cap _{g \in G}}{I^g}\, = \,P\, \cap \,K[N]$.


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DOI: https://doi.org/10.1090/S0002-9947-1979-0539920-X
Keywords: Group ring, irreducible module, primitive ideal, maximal ideal, Ore extension, group acting on a group algebra, classical ring of quotients, Jacobson topology
Article copyright: © Copyright 1979 American Mathematical Society