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 $\sigma$.
@article{VMJ_2016_18_3_a3,
author = {R. Y. Dryaeva and V. A. Koibaev},
title = {An elementary net associated with the elementary group},
journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
pages = {31--34},
year = {2016},
volume = {18},
number = {3},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/VMJ_2016_18_3_a3/}
}
TY - JOUR
AU - R. Y. Dryaeva
AU - V. A. Koibaev
TI - An elementary net associated with the elementary group
JO - Vladikavkazskij matematičeskij žurnal
PY - 2016
SP - 31
EP - 34
VL - 18
IS - 3
UR - http://geodesic.mathdoc.fr/item/VMJ_2016_18_3_a3/
LA - ru
ID - VMJ_2016_18_3_a3
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%A V. A. Koibaev
%T An elementary net associated with the elementary group
%J Vladikavkazskij matematičeskij žurnal
%D 2016
%P 31-34
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%N 3
%U http://geodesic.mathdoc.fr/item/VMJ_2016_18_3_a3/
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%F VMJ_2016_18_3_a3
R. Y. Dryaeva; V. A. Koibaev. An elementary net associated with the elementary group. Vladikavkazskij matematičeskij žurnal, Tome 18 (2016) no. 3, pp. 31-34. http://geodesic.mathdoc.fr/item/VMJ_2016_18_3_a3/
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[6] Koibaev V. A., Nuzhin Ya. N., “Podgruppy grupp Shevalle i koltsa Li, opredelyaemye naborom additivnykh podgrupp osnovnogo koltsa”, Fundament. i prikl. matem., 18:1 (2013), 75–84 | MR
