Decomposition of Elementary Transvection in Elementary Net Group

Itarova, S. Y. , Koibaev, V. A.
Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 3.

Abstract:
The paper deals with the study of elementary nets (carpets) \(\sigma = (\sigma_{ij})\) and elementary net groups \(E(\sigma)\). Namely, decomposition of an elementary transvection in elementary net group \(E(\sigma)\) is given. The colections of subsets (ideals, additive subgroups and etc.) \( \sigma=\{\sigma_{ij}: 1\leq i, j\leq n\}\) of an associative ring with the conditions \(\sigma_{ir}\sigma_{rj}\subseteq\sigma_{ij}\), \(1\leq i,r,j\leq n,\) arose in a different situations. Such collections are called carpets or nets and a rings, while the associated groups are called carpet (net, congruence, etc.) subgroups. An elementary net (a net without diagonal) \(\sigma\) is closed (admissible) if the subgroup \(E(\sigma)\) does not contain new elementary transvections. The study was motivated by the question of V. M. Levchuk (The Kourovka notebook, question 15.46) whether or not a necessary and sufficient condition for the admissibility (closure) of the elementary net \(\sigma\) is the admissibility (closure) of all pairs \((\sigma_{ij}, \sigma_{ji})\). In other words, the inclusion of an elementary transvection \(t_{ij}(\alpha)\) in the elementary group \(E(\sigma)\) is equivalent to the inclusion of \(t_{ij}(\alpha)\) in the subgroup \(\langle t_{ij}(\sigma_{ij}), t_{ji}(\sigma_{ji}) \rangle\) (for any \(i\neq j\)). Thus, the decomposition of elementary transvection \(t_{ij}(\alpha)\) in the elementary net group \(E(\sigma)\) becomes relevant. We consider an elementary net \(\sigma=(\sigma_{ij})\) (elementary carpet) of the additive subgroups of a commutative ring of order \(n\), a derived net \(\omega=(\omega_{ij})\) depending on the net \(\sigma\), the net \(\Omega=(\Omega_{ij})\) associated with the elementary group \(E(\sigma)\), where \(\omega\subseteq\sigma\subseteq\Omega\) and the net \(\Omega\) is the least (complemented) net among all the nets which contain the elementary net \(\sigma\). Let \(R\) be a commutative unital ring and \(n\in\Bbb{N}\), \(n\geq 2\). A set \( \sigma = (\sigma_{ij})\), \(1\leq{i, j} \leq{n},\) of additive subgroups \(\sigma_{ij}\) of the ring \(R\) is said to be a net or a carpet over the ring \(R\) of order \(n\) if \(\sigma_{ir} \sigma_{rj} \subseteq{\sigma_{ij}}\) for all \(i\), \(r\), \(j\). A net without diagonal is said to be  elementary net or  elementary carpet. We prove that every elementary transvection \(t_{ij}(\alpha)\in E(\sigma)\) can be decomposed \(t_{ij}(\alpha)=ah\) into a product of two matrices \(a\) and \(h\), where \(a\) is a member of the group \(\langle t_{ij}(\sigma_{ij}),t_{ji}(\sigma_{ji})\rangle\), \(h\) is a member of the net group \(G(\tau)\), where \(\tau =\begin{pmatrix} \tau_{ii} & \omega_{ij}  \omega_{ji} & \tau_{jj} \end{pmatrix}\), \(\omega_{ii}\subseteq \tau_{ii} \subseteq \Omega_{ii}\). Important characteristics of matrices \(a\) and \(h\) involved in the decomposition of elementary transvection \(t_{ij}(\alpha)\) were also obtained in the paper.

Keywords: nets, carpets, elementary net, net group, closed net, derivative net, elementary net group, transvections.

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