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**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.

Reviewer: János Kurdics (Nyíregyháza)

### 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 |

### Keywords:

torsion-complete groups; modular Abelian group rings; groups of units; bounded Abelian groups; \(p\)-reduced groups; \(p\)-mixed groups; normalized units; Cauchy sequences; nilpotent elements### Citations:

Zbl 1106.16034
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\textit{P. Danchev}, Period. Math. Hung. 59, No. 1, 37--42 (2009; Zbl 1196.16032)

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### 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 |

[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 |

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