×

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

Citations:

Zbl 1106.16034
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] P. V. Danchev, Torsion completeness of Sylow p-groups in modular group rings, Acta Math. Hungar., 75 (1997), 317–322. · Zbl 0927.16031 · doi:10.1023/A:1006597605945
[2] P. V. Danchev, Quasi-completeness in commutative modular group algebras, Ricerche Mat., 51 (2002), 319–327. · Zbl 1141.16306
[3] P. V. Danchev, Quasi-closed primary components in abelian group rings, Tamkang J. Math., 34 (2003), 87–92. · Zbl 1035.16026
[4] P. V. Danchev, Torsion-complete primary components in modular abelian group rings over certain rings, Sci. Ser. A Math. Sci. (N.S.), 12 (2006), 17–20. · Zbl 1106.16034
[5] P. V. Danchev, On the balanced subgroups of modular abelian group rings, Vladikavkaz. Mat. Zh., 8 (2006), 29–32. · Zbl 1289.16073
[6] L. Fuchs, Infinite Abelian Groups, Vol. I, II, Mir, Moskva, 1974, 1977 (in Russian). · Zbl 0274.20067
[7] N. A. Nachev, Torsion-completeness of the group of normalized units in modular group rings, C. R. Acad. Bulgare Sci., 47 (1994), 9–11. · Zbl 0823.16022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.