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.


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
Full Text: DOI


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