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Vladikavkazskii Matematicheskii Zhurnal, 2007, Volume 9, Number 2, Pages 3–8
(Mi vmj90)
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This article is cited in 2 scientific papers (total in 2 papers)
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.
Received: 03.07.2006
Citation:
P. V. Danchev, “On a decomposition equality in modular group rings”, Vladikavkaz. Mat. Zh., 9:2 (2007), 3–8
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https://www.mathnet.ru/eng/vmj90 https://www.mathnet.ru/eng/vmj/v9/i2/p3
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Abstract page: | 265 | Full-text PDF : | 120 | References: | 54 | First page: | 1 |
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