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DOI: 10.23671/VNC.2017.1.5818

Cyclical Elementary Nets

Dzhusoeva, N. A. , Dryaeva R. Y.
Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 1.
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 odd-order 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, non-split maximal torus, net, cyclic net, net group, elementary group, transvection.
Language: Russian Download the full text  
For citation: Dzhusoeva N. A., Dryaeva R. Y. Cyclical elementary nets. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp.26-29. DOI 10.23671/VNC.2017.1.5818
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