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Zapiski Nauchnykh Seminarov POMI, 2015, Volume 435, Pages 33–41
(Mi znsl6149)
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This article is cited in 5 scientific papers (total in 5 papers)
Decomposition of elementary transvection in elementary group
R. Yu. Dryaevaa, V. A. Koibaevab a North Ossetian State University after Kosta Levanovich Khetagurov, Vladikavkaz, Russia
b Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, Vladikavkaz, Russia
Abstract:
We consider the following data: an elementary net (or, what is the same elementary carpet) $\sigma=\sigma_{ij})$ of additive subgroups of a commutative ring (in other words, a net without the diagonal) of order $n$, a derived net $\omega=(\omega_{ij})$, which depends of 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 smallest (complemented) net among the all nets which contain the elementary net $\sigma$. We prove that every elementary transvection $t_{ij}(\alpha)$ can be decomposed as a product of two matrices $M_1$ and $M_2$, where $M_1$ belongs to the group $\langle t_{ij}\sigma_{ij}),t_{ji}(\sigma_{ji})\rangle$, $M_2$ belongs to the net group $G(\tau)$ and the net $\tau$ has the form $\tau=\begin{pmatrix}\Omega_{11}&\omega_{12}\\\omega_{21}&\Omega_{22}\end{pmatrix}$.
Key words and phrases:
nets, elementary nets, closed nets, net groups, elementary group, transvection.
Received: 23.09.2015
Citation:
R. Yu. Dryaeva, V. A. Koibaev, “Decomposition of elementary transvection in elementary group”, Problems in the theory of representations of algebras and groups. Part 28, Zap. Nauchn. Sem. POMI, 435, POMI, St. Petersburg, 2015, 33–41; J. Math. Sci. (N. Y.), 219:4 (2016), 513–518
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https://www.mathnet.ru/eng/znsl6149 https://www.mathnet.ru/eng/znsl/v435/p33
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Abstract page: | 309 | Full-text PDF : | 56 | References: | 69 |
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