Skip to Main Content

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 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Primitive group rings and Noetherian rings of quotients
HTML articles powered by AMS MathViewer

by Christopher J. B. Brookes and Kenneth A. Brown
Trans. Amer. Math. Soc. 288 (1985), 605-623
DOI: https://doi.org/10.1090/S0002-9947-1985-0776395-2

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 $A \cap Z = 1$.
References
Similar Articles
Bibliographic 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