## 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

### MSC:

 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