Torsion completeness of $$p$$-primary components in modular group rings of $$p$$-reduced Abelian groups.(English)Zbl 1196.16032

Let $$G$$ be an Abelian group, $$R$$ a commutative ring with unity of prime characteristic $$p$$, $$RG$$ the group ring, $$S$$ the normalized $$p$$-Sylow subgroup of its group of units. The author, strengthening his earlier result [in Sci., Ser. A, Math. Sci. (N.S.) 12, 17-20 (2006; Zbl 1106.16034)], proves that if $$G$$ is a $$p$$-reduced group then the group $$S$$ is torsion-complete if and only if there exists a natural number $$i$$ such that either $$R^{p^i}$$ has no nontrivial nilpotent elements and the $$p$$-component $$G_p$$ is bounded, or $$R^{p^i}$$ has a nontrivial nilpotent element for every natural number $$i$$ and $$G$$ is a bounded $$p$$-group. Also, the following conjecture is stated: if the nontrivial group $$G$$ is $$p$$-divisible and the radical $$N(R^{p^i})\neq 0$$ for every natural number $$i$$ then the group $$S$$ is not torsion-complete.

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 20K21 Mixed groups

Zbl 1106.16034
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References:

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