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 Vladikavkaz. Mat. Zh., 2007, Volume 9, Number 2, Pages 3–8 (Mi vmj90)

On a decomposition equality in modular group rings

P. V. Danchev

Plovdiv State University «Paissii Hilendarski», Plovdiv, Bulgaria

Abstract: Let $G$ be an abelian group such that $A\le G$ with $p$-component $A_p$ and $B\le G$, and let $R$ be a commutative ring with 1 of prime characteristic $p$ with nil-radical $N(R)$. It is proved that if $A_p\not\subseteq B_p$ or $N(R)\ne 0$, then $S(RG)=S(RA)(1+I_p(RG;B))$ $\iff$ $G=AB$ and $G_p=A_pB_p$. In particular, if $A_p\ne 1$ or $N(R)\ne 0$, then $S(RG)=S(RA)\times (1+I_p(RG;B))$ $\iff$ $G=A\times B$. So, the question concerning the validity of this formula is completely exhausted. The main statement encompasses both the results of this type established by the author in (Hokkaido Math. J., 2000) and (Miskolc Math. Notes, 2005). We also point out and eliminate in a concrete situation an error in the proof of a statement due to T. Zh. Mollov on a decomposition formula in commutative modular group rings (Proceedings of the Plovdiv University-Math., 1973).

Key words: direct factors, decompositions, normed unit groups, homomorphisms.

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Bibliographic databases:
UDC: 512.742
MSC: 16S34, 16U60, 20K10, 20K21
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Citation: P. V. Danchev, “On a decomposition equality in modular group rings”, Vladikavkaz. Mat. Zh., 9:2 (2007), 3–8

Citation in format AMSBIB
\Bibitem{Dan07} \by P.~V.~Danchev \paper On a decomposition equality in modular group rings \jour Vladikavkaz. Mat. Zh. \yr 2007 \vol 9 \issue 2 \pages 3--8 \mathnet{http://mi.mathnet.ru/vmj90} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2434626} 

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This publication is cited in the following articles:
1. P. V. Danchev, “A note on decompositions in abelian group rings”, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat., 16:1 (2008), 73–76
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