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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Primitive group rings and Noetherian rings of quotients
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by Christopher J. B. Brookes and Kenneth A. Brown PDF
Trans. Amer. Math. Soc. 288 (1985), 605-623 Request permission

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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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 288 (1985), 605-623
  • MSC: Primary 16A27; Secondary 16A20, 16A33, 20C07
  • DOI: https://doi.org/10.1090/S0002-9947-1985-0776395-2
  • MathSciNet review: 776395