Transactions of the American Mathematical Society

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



Primitive group rings and Noetherian rings of quotients

Authors: Christopher J. B. Brookes and Kenneth A. Brown
Journal: Trans. Amer. Math. Soc. 288 (1985), 605-623
MSC: Primary 16A27; Secondary 16A20, 16A33, 20C07
MathSciNet review: 776395
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Abstract: Let $ k$ be a field, and let $ G$ be a countable nilpotent group with centre $ Z$. We show that the group algebra $ kG$ is primitive if and only if $ k$ is countable, $ G$ is torsion free, and there exists an abelian subgroup $ A$ of $ G$, of infinite rank, with $ A \cap Z = 1$. Suppose now that $ G$ is torsion free. Then $ kG$ has a partial quotient ring $ Q = kG{(kZ)^{ - 1}}$. The above characterisation of the primitivity of $ kG$ is intimately connected with the question: When is $ Q$ a Noetherian ring? We determine this for those groups $ G$, as above, all of whose finite rank subgroups are finitely generated. In this case, $ Q$ is Noetherian if and only if $ G$ has no abelian subgroup $ A$ of infinite rank with $ A \cap Z = 1$.

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Keywords: Group ring, primitive ring, Noetherian ring, nilpotent group
Article copyright: © Copyright 1985 American Mathematical Society