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DOI: 10.23671/VNC.2016.3.5874

# An Elementary Net Associated with the Elementary Group

Dryaeva R. Y. , Koibaev, V. A.
Vladikavkaz Mathematical Journal 2016. Vol. 18. Issue 3.
Abstract:
Let $$R$$ be an arbitrary commutative ring with identity, $$n$$ be a positive integer, $$n\geq 2$$. The set  $$\sigma = (\sigma_{ij})$$, $$1 \leq {i, j} \leq {n},$$ of additive subgroups of the ring $$R$$ is called a net (or {\it 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 {\it 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 $$\sigma$$.
Keywords: carpet, elementary carpet, net, elementary net, net group, elementary group, transvection