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Trivial units in Abelian group algebras. (English) Zbl 1184.16040

Summary: We find a necessary and sufficient condition when all units in arbitrary Abelian group algebras are trivial by reducing it to the torsion case. This is a supplement to our recent results [in Extr. Math. 23, No. 1, 49-60 (2008; Zbl 1163.16019)]. In particular, we extend theorems due to J. Ritter, S. K. Sehgal [Math. Proc. R. Ir. Acad. 105A, No. 1, 25-39 (2005; Zbl 1101.16024)] and A. Herman, Y. Li [Proc. Am. Math. Soc. 134, No. 3, 631-635 (2006; Zbl 1087.16017)].

MSC:

16U60 Units, groups of units (associative rings and algebras)
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16S34 Group rings
20K21 Mixed groups
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