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Torsion completeness of Sylow $$p$$-groups in modular group rings. (English) Zbl 0927.16031
Summary: Let $$R$$ be commutative ring with identity of prime characteristic $$p$$, let $$N(R)=\text{rad }R$$ be the nilradical (a Baer radical) in $$R$$, and let $$G$$ be an arbitrary Abelian group written multiplicatively with $$p$$-primary component $$G_p$$. $$RG$$ denotes an Abelian group ring of $$G$$ over $$R$$, and $$S(RG)$$ is its normed Sylow $$p$$-group.
In the present paper a criterion is given for $$S(RG)$$ to be torsion complete when $$R$$ is a ring with nilpotent elements, i.e., when $$R$$ possesses a nontrivial nilradical $$N(R)\neq 0$$. The first main result of this paper is that $$S(RG)$$ is torsion complete if and only if $$G$$ is bounded $$p$$-torsion provided $$R$$ is perfect. The next major result is a description of torsion completeness of $$S(RG)$$ provided $$R$$ is a weakly perfect ring.

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 20K10 Torsion groups, primary groups and generalized primary groups
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