Geodesic
On a decomposition equality in modular group rings
Vladikavkazskij matematičeskij žurnal, Tome 9 (2007) no. 2, pp. 3-8 
P. V. Danchev. On a decomposition equality in modular group rings. Vladikavkazskij matematičeskij žurnal, Tome 9 (2007) no. 2, pp. 3-8. http://geodesic.mathdoc.fr/item/VMJ_2007_9_2_a0/
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     title = {On a decomposition equality in modular group rings},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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).

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