An Embedding Theorem for an Elementary Net

Dzhusoeva, N. A. , Itarova S. Y. , Koibaev, V. A.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 2.

Abstract:
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 \(\omega=(\omega_{ij})\) ic completed with a diagonal to the full net in the second way then the inclusions are valid for all \(i\), \(r\), \(j\).

Keywords: nets, elementary nets, net groups, derivative nets, elementary net groups, transvections

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