Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 16A27, 16A20, 16A33, 20C07

Retrieve articles in all journals with MSC: 16A27, 16A20, 16A33, 20C07


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0776395-2
PII: S 0002-9947(1985)0776395-2
Keywords: Group ring, primitive ring, Noetherian ring, nilpotent group
Article copyright: © Copyright 1985 American Mathematical Society