Cyclical elementary nets

N. A. Dzhusoeva ; R. Y. Dryaeva

Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 1, pp. 26-29

Résumé

Let $R$ be a commutative ring with the unit and $n\in\mathbb{N}$. A set $\sigma = (\sigma_{ij})$, $1\leqslant{i, j} \leqslant{n},$ 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 $1\leqslant i, r, j\leqslant n$. A net which doesn't contain the diagonal is called an elementary net. An elementary net $\sigma = (\sigma_{ij}), 1\leqslant{i\neq{j} \leqslant{n}}$, is complemented, if for some additive subgroups $\sigma_{ii}$ of $R$ the set $\sigma = (\sigma_{ij}), 1\leqslant{i, j} \leqslant{n}$ 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\leqslant{i\neq{j}} \leqslant{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 $n\in\mathbb{N}$ there exists an elementary cyclic net which is not complemented.