Decomposition of elementary transvection in elementary net group

S. Yu. Itarova ; V. A. Koibaev

Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 3, pp. 24-30

Résumé

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.