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.


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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