Units of commutative group algebras. (English) Zbl 0703.16017

The paper, dedicated to the memory of the famous Soviet mathematician S. D. Berman (1922-1987), is a review of some results on units of group algebras. Let \(\mathcal{RG}\) be the group algebra of an Abelian group \(\mathcal G\) over a commutative ring \(\mathcal R\), \(\mathcal U(\mathcal{RG})\) be the unit group of \(\mathcal{RG}\) and \(V(\mathcal{RG})\) be the group of all normalized units of \(RG\). The paper presents (a) a description of the idempotent subgroup of \(\mathcal V(\mathcal{RG})\) (b) a description of \(\mathcal U(\mathcal{RG})\) when the group \(\mathcal G\) is torsion-free and (c) necessary and sufficient conditions for the group \(\mathcal U(\mathcal{RG})\) to be finitely generated and the group \(\mathcal G\) to be a direct factor of \(\mathcal V(\mathcal{RG})\). The isomorphism class of \(\mathcal V(\mathcal{RG})\) is determined when \(\mathcal G\) is an Abelian \(p\)-group and \(\mathcal R\) is a field of characteristic \(p\). The structure of \(\mathcal U(\mathbb{Z}\mathcal G)\) is given as well as a result of Bass about a specific fundamental system of units of \(\mathbb{Z}\mathcal G\).
Reviewer: T.Mollov


16U60 Units, groups of units (associative rings and algebras)
16S34 Group rings
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20K10 Torsion groups, primary groups and generalized primary groups