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DOI: 10.23671/VNC.2018.2.14721

# 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
Language: Russian Download the full text For citation: Dzhusoeva N. A., Itapova C. Y., Koibaev V. A. An embedding theorem for an elementary net. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no. 2, pp.57-61. DOI 10.23671/VNC.2018.2.14721

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