Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- An army of cohomology against residual finiteness
- On some questions concerning subnormally monomial groups
- A conjecture concerning the evaluation of products of class-sums of the symmetric group
- Automorphisms of Burnside rings
- On finite generation of unit groups for group rings
- Counting finite index subgroups
- The quantum double of a finite group and its role in conformal field theory
- Closure properties of supersoluble Fitting classes
- Groups acting on locally finite graphs - a survey of the infinitely ended case
- An invitation to computational group theory
- On subgroups, transversals and commutators
- Intervals in subgroup lattices of finite groups
- Amalgams of minimal local subgroups and sporadic simple groups
- Vanishing orbit sums in group algebras of p-groups
- From stable equivalences to Rickard equivalences for blocks with cyclic defect
- Factorizations in which the factors have relatively prime orders
- Some problems and results in the theory of pro-p groups
- On equations in finite groups and invariants of subgroups
- Group presentations where the relators are proper powers
- A condensing theorem
- Lie methods in group theory
- Some new results on arithmetical problems in the theory of finite groups
- Groups that admit partial power automorphisms
- Problems
On finite generation of unit groups for group rings
Published online by Cambridge University Press: 19 February 2010
- Frontmatter
- Contents
- Preface
- Introduction
- An army of cohomology against residual finiteness
- On some questions concerning subnormally monomial groups
- A conjecture concerning the evaluation of products of class-sums of the symmetric group
- Automorphisms of Burnside rings
- On finite generation of unit groups for group rings
- Counting finite index subgroups
- The quantum double of a finite group and its role in conformal field theory
- Closure properties of supersoluble Fitting classes
- Groups acting on locally finite graphs - a survey of the infinitely ended case
- An invitation to computational group theory
- On subgroups, transversals and commutators
- Intervals in subgroup lattices of finite groups
- Amalgams of minimal local subgroups and sporadic simple groups
- Vanishing orbit sums in group algebras of p-groups
- From stable equivalences to Rickard equivalences for blocks with cyclic defect
- Factorizations in which the factors have relatively prime orders
- Some problems and results in the theory of pro-p groups
- On equations in finite groups and invariants of subgroups
- Group presentations where the relators are proper powers
- A condensing theorem
- Lie methods in group theory
- Some new results on arithmetical problems in the theory of finite groups
- Groups that admit partial power automorphisms
- Problems
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 : b ∈ B} for any subset B ⊂ A. 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.
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- Chapter
- Information
- Groups '93 Galway/St Andrews , pp. 352 - 367Publisher: Cambridge University PressPrint publication year: 1995
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