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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.
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 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.