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DOI: 10.23671/VNC.2017.1.5818 Cyclical Elementary Nets
Abstract:
Let \(R\) be a commutative ring with the unit and \(n\in\Bbb{N}\). A set \(\sigma = (\sigma_{ij})\), \(1\leq{i, j} \leq{n},\) of additive subgroups of the ring \(R\) is a net over \(R\) of order \(n\), if \( \sigma_{ir} \sigma_{rj} \subseteq{\sigma_{ij}} \) for all \(1\leq i, r, j\leq n\). A net which doesn't contain the diagonal is called an elementary net. An elementary net \(\sigma = (\sigma_{ij}), 1\leq{i\neq{j} \leq{n}}\), is complemented, if for some additive subgroups \(\sigma_{ii}\) of \(R\) the set \(\sigma = (\sigma_{ij}), 1\leq{i, j} \leq{n}\) is a full net. An elementary net \(\sigma\) is called closed, if the elementary group \( E(\sigma) = \langle t_{ij}(\alpha) : \alpha\in \sigma_{ij}, 1\leq{i\neq{j}} \leq{n}\rangle \) doesn't contain elementary transvections. It is proved that the cyclic elementary oddorder nets are complemented. In particular, all such nets are closed. It is also shown that for every odd \(n\in\Bbb{N}\) there exists an elementary cyclic net which is not complemented.
Keywords: intermediate subgroup, nonsplit maximal torus, net, cyclic net, net group, elementary group, transvection.
Language: Russian
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For citation: Dzhusoeva N. A., Dryaeva R. Y. Cyclical elementary nets. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp.2629. DOI 10.23671/VNC.2017.1.5818 ← Contents of issue 
 

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