An elementary net associated with the elementary group

R. Y. Dryaeva ; V. A. Koibaev

Vladikavkazskij matematičeskij žurnal, Tome 18 (2016) no. 3, pp. 31-34

Résumé

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 $\sigma$.