An embedding theorem for an elementary net

N. A. Dzhusoeva ; S. Yu. Itarova ; V. A. Koibaev

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 2, pp. 57-61

Résumé

Let $\Lambda$ be a commutative unital ring and $n\in\Bbb{N}$, $n\geq 2$. A set $\sigma = (\sigma_{ij})$, $1\leq{i, j} \leq{n}, $ of additive subgroups $\sigma_{ij}$ of $\Lambda$ is said to be a net or a carpet of order $n$ over the ring $\Lambda$ if $\sigma_{ir} \sigma_{rj} \subseteq{\sigma_{ij}}$ for all $i$, $r$, $j$. A net without diagonal is called an elementary net. An elementary net $\sigma=(\sigma_{ij})$, $1\leq{i\neq{j} \leq{n}}$, is said to be complemented (to a full net), if for some additive subgroups (subrings) $\sigma_{ii}$ of $\Lambda$ the matrix (with the diagonal) $\sigma = (\sigma_{ij})$, $1\leq{i,j}\leq{n}$ is a full net. Assume that $\sigma = (\sigma_{ij})$ is an elementary net over the ring $\Lambda$ of the order $n$. Consider a set $\omega = (\omega_{ij})$ of additive subgroups $\omega_{ij}$ of the ring $\Lambda$, where $i\neq{j}$ defined by the rule $\omega_{ij}= \sum_{k=1}^{n}\sigma_{ik}\sigma_{kj},$ $k\neq i;\ k\neq j$. The set $\omega = (\omega_{ij})$ of elementary subgroups $\omega_{ij}$ of the ring $\Lambda$ is an elementary net called an elementary derived net. An elementary net $\omega$ can be completed to a full net by the standard way. In this article we propose a second way to complete an elementary net to a full net. The notion of a net $\Omega=(\Omega_{ij})$ associated with an elementary group $E(\sigma)$ is also introduced. The following theorem is the main result of the paper: An elementary net $\sigma$ generates an elementary derived net $\omega=(\omega_{ij})$ and a net $\Omega=(\Omega_{ij})$ associated with the elementary group $E(\sigma)$ such that $\omega\subseteq \sigma \subseteq \Omega$. If $\omega=(\omega_{ij})$ is completed with a diagonal to the full net in the standard way, then for all $r$ and $i\neq j$ we have $\omega_{ir}\Omega_{rj} \subseteq \omega_{ij}$ and $\Omega_{ir}\omega_{rj} \subseteq \omega_{ij}$. If $\omega=(\omega_{ij})$ is completed with a diagonal to the full net in the second way then the inclusions are valid for all $i$, $r$, $j$.