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Idempotent-nilpotent units in commutative group rings. (English) Zbl 1232.16024

Let \(G\) be a nontrivial Abelian group, \(R\) a commutative ring with unity of prime characteristic \(p\), \(RG\) the group algebra, \(V(RG)\) the group of normalized units. Completing his research exposed in several research papers [such as Kochi Math. J. 4, 61-66 (2009; Zbl 1182.16031)], the author characterises group rings \(RG\) for which the decomposition \(V(RG)=Id(RG)\times 1+I(N(R)G)\) holds, where \(Id(RG)\) is the group of idempotent units, \(N(R)\) is the nilradical. Namely, this happens in case \(G\) is torsion-free, the classical case due to G. Karpilovsky [Expo. Math. 8, No. 3, 247-287 (1990; Zbl 0703.16017)], and in three additional cases with \(G\) of order 2 or 3.

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
20K20 Torsion-free groups, infinite rank
20K21 Mixed groups
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