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

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