Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-07-01T15:06:46.146Z Has data issue: false hasContentIssue false

On finite generation of unit groups for group rings

Published online by Cambridge University Press:  19 February 2010

C. M. Campbell
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University College, Galway
T. C. Hurley
Affiliation:
University of St Andrews, Scotland
S. J. Tobin
Affiliation:
University College, Galway
J Krempa
Affiliation:
Institute of Mathematics, Warsaw University, ul. Banacha 2, 02–097 Warszawa, Poland
Get access

Summary

Abstract

Let R represent an associative, but nonnecessarily commutative ring with 1 ≠ 0, G a nontrivial group, and RG the group ring of G over R.

Let us consider the following problem: Find the necessary and sufficient conditions under which the unit group of RG, or the group of normalized units of RG, is finitely generated.

We are going to survey and extend some known results about this problem. We also formulate several more detailed questions suggested by this survey.

Preliminaries

In this paper we assume that all rings are associative with 1 ≠ 0. Subrings with the same unities will be called unital. For convenience of readers we recall some notation and terminology from ring theory.

U(A) will always denote the unit group of the ring A, A+ the additive group of A, and 1 + B – the set {1 + b : bB} for any subset BA. Let us also agree that J(A) will stand for the Jacobson radical of the ring A, and N(A) for the set of all nilpotent elements of A. Further we will say that a ring A is semisimple if J(A) = 0, and reduced if N(A) = 0. Clearly if A is commutative then N(A) is an ideal contained in J(A) and the factor ring A/N(A) is reduced. Rings having no proper central idempotents will be called here indecomposable.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×