Geodesic
Decomposition of elementary transvection in elementary group
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 28, Tome 435 (2015), pp. 33-41 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\tau=\begin{pmatrix}\Omega_{11}&\omega_{12}\\\omega_{21}&\Omega_{22}\end{pmatrix}$. 
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     author = {R. Yu. Dryaeva and V. A. Koibaev},
     title = {Decomposition of elementary transvection in elementary group},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {33--41},
     year = {2015},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_435_a1/}
}
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R. Yu. Dryaeva; V. A. Koibaev. Decomposition of elementary transvection in elementary group. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 28, Tome 435 (2015), pp. 33-41. http://geodesic.mathdoc.fr/item/ZNSL_2015_435_a1/

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