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.