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 Vladikavkaz. Mat. Zh., 2017, Volume 19, Number 1, Pages 26–29 (Mi vmj604)

Cyclical elementary nets

N. A. Dzhusoeva, R. Y. Dryaeva

North Ossetian State University after Kosta Levanovich Khetagurov, Vladikavkaz

Abstract: Let $R$ be a commutative ring with the unit and $n\in\mathbb{N}$. A set $\sigma = (\sigma_{ij})$, $1\leqslant{i, j} \leqslant{n},$ of additive subgroups of the ring $R$ is a net over $R$ of order $n$, if $\sigma_{ir} \sigma_{rj} \subseteq{\sigma_{ij}}$ for all $1\leqslant i, r, j\leqslant n$. A net which doesn't contain the diagonal is called an elementary net. An elementary net $\sigma = (\sigma_{ij}), 1\leqslant{i\neq{j} \leqslant{n}}$, is complemented, if for some additive subgroups $\sigma_{ii}$ of $R$ the set $\sigma = (\sigma_{ij}), 1\leqslant{i, j} \leqslant{n}$ is a full net. An elementary net $\sigma$ is called closed, if the elementary group $E(\sigma) = \langle t_{ij}(\alpha) : \alpha\in \sigma_{ij}, 1\leqslant{i\neq{j}} \leqslant{n}\rangle$ doesn't contain elementary transvections. It is proved that the cyclic elementary odd-order nets are complemented. In particular, all such nets are closed. It is also shown that for every odd $n\in\mathbb{N}$ there exists an elementary cyclic net which is not complemented.

Key words: intermediate subgroup, non-split maximal torus, net, cyclic net, net group, elementary group, transvection.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 115033020013

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Citation: N. A. Dzhusoeva, R. Y. Dryaeva, “Cyclical elementary nets”, Vladikavkaz. Mat. Zh., 19:1 (2017), 26–29

Citation in format AMSBIB
\Bibitem{DzhDry17} \by N.~A.~Dzhusoeva, R.~Y.~Dryaeva \paper Cyclical elementary nets \jour Vladikavkaz. Mat. Zh. \yr 2017 \vol 19 \issue 1 \pages 26--29 \mathnet{http://mi.mathnet.ru/vmj604}