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 Vladikavkaz. Mat. Zh., 2016, Volume 18, Number 3, Pages 31–34 (Mi vmj587)

An elementary net associated with the elementary group

R. Y. Dryaevaa, V. A. Koibaevab

a North Ossetian State University after Kosta Levanovich Khetagurov, Vladikavkaz, Russia
b Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, Vladikavkaz, Russia

Abstract: Let $R$ be an arbitrary commutative ring with identity, $n$ be a positive integer, $n\geq2$. The set $\sigma=(\sigma_{ij})$, $1\leq i,j\leq n$, of additive subgroups of the ring $R$ is called a net (or carpet) over the ring $R$ of order $n$, if the inclusions $\sigma_{ir}\sigma_ {rj}\subseteq\sigma_{ij}$ hold for all $i,r,j$. The net without the diagonal, is called an elementary net. The elementary net $\sigma=(\sigma_{ij})$, $1\leq i\neq j\leq n$, is called complemented, if for some additive subgroups $\sigma_{ii}$ of the ring $R$ the set $\sigma=(\sigma_ {ij})$, $1\leq i,j\leq n$ is a (full) net. The elementary net $\sigma=(\sigma_{ij})$ is complemented if and only if the inclusions $\sigma_{ij}\sigma_{ji}\sigma_{ij}\subseteq\sigma_{ij}$ hold for any $i\neq j$. Some examples of not complemented elementary nets are well known. With every net $\sigma$ can be associated a group $G(\sigma)$ called a net group. This groups are important for the investigation of different classes of groups.
It is proved in this work that for every elementary net $\sigma$ there exists another elementary net $\Omega$ associated with the elementary group $E(\sigma)$. It is also proved that an elementary net $\Omega$ associated with the elementary group $E(\sigma)$ is the smallest elementary net that contains the elementary net $\sigma$.

Key words: carpet, elementary carpet, net, elementary 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: R. Y. Dryaeva, V. A. Koibaev, “An elementary net associated with the elementary group”, Vladikavkaz. Mat. Zh., 18:3 (2016), 31–34

Citation in format AMSBIB
\Bibitem{DryKoi16} \by R.~Y.~Dryaeva, V.~A.~Koibaev \paper An elementary net associated with the elementary group \jour Vladikavkaz. Mat. Zh. \yr 2016 \vol 18 \issue 3 \pages 31--34 \mathnet{http://mi.mathnet.ru/vmj587}