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Published online by Cambridge University Press: 04 August 2010
Abstract
In this note we will survey and extend some results on periodicity of unit groups of associative rings. Special attention will be paid to group rings.
Preliminaries
In this paper we assume that rings are associative, in general with 1 ≠ 0. If R is a ring then U(R) will denote the unit group of the ring R, R+ the additive group of R, RU the subring of R generated by U(R) and J(R) the (Jacobson) radical of the ring R. By an order we mean here a ℤ-order. For other notions and results of ring theory one can consult for example [17].
We will apply rather standard notation and terminology on groups. For example, Cn will denote the cyclic group of order n and Q8 the quaternion group of order 8. For further information about groups see for example [18, 23].
Various finiteness conditions for groups of units of associative rings, in particular of group rings are studied in the literature (see for example [25, 20, 13, 26, 15]). In this paper we are going to concentrate on periodicity of groups of units.
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