The set $\sigma=(\sigma_{ij})$, $1\leq{i, j}\leq{n},$ of additive subgroups $\sigma_{ij}$ of a field $K$ is called a net (carpet) over $K$ of order $n$ if $\sigma_{ir} \sigma_{rj} \subseteq{\sigma_{ij}}$ for all values of the indices $i$, $r$, and $j$. A net considered without the diagonal is called an elementary net. Based on an elementary net $\sigma$, an elementary net subgroup $E(\sigma)$ is defined, which is generated by elementary transvections $t_{ij}(\alpha) = e+\alpha e_{ij}$. An elementary net $\sigma$ is called closed if the subgroup $E(\sigma)$ does not contain new elementary transvections. Suppose that $R$ is a Noetherian domain with the QR-property (i.e., any intermediate subring lying between $R$ and its field of fractions $K$ is a ring of fractions of the ring $R$ with respect to a multiplicative system in $R$), $\sigma=(\sigma_{ij})$ is a complete (elementary) net of order $n\geq 2$ ($n\geq 3$, respectively) over $K$, and the additive subgroups $\sigma_{ij}$ are nonzero $R$-modules. It is proved that, up to conjugation by a diagonal matrix, all $\sigma_{ij}$ are (fractional) ideals of a fixed intermediate subring $P$, $R\subseteq P \subseteq K$, and the inclusions $\pi_{ij}\pi_{ji}\subseteq P$ and $\pi_{ij}\subseteq P\subseteq\pi_{j i}$ hold for all $i<j$. In particular, the elementary net $\sigma$ is closed.
Keywords: general and special linear groups, full and elementary nets (carpets) of additive subgroups, net subgroup
Received January 23, 2024
Revised August 24, 2024
Accepted September 2, 2024
Funding Agency: The work was performed with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2024-1447).
Roksana Yurievna Dryaeva, North-Ossetian State University named after K. L. Khetagurov, Vladikavkaz, 362025 Russia, e-mail: dryaeva-roksana@mail.ru
Vladimir Amurkhanovich Koibaev, Dr. Phys.-Math. Sci., Prof., North-Ossetian State University named after K. L. Khetagurov, Vladikavkaz, 362025 Russia; Southern Mathematical Institute VSC RAS, Vladikavkaz, 362025 Russia, e-mail: koibaev-K1@yandex.ru
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Cite this article as: R.Yu. Dryaeva, V.A. Koibaev. Full and elementary nets over the field of fractions of a ring with the QR-property. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 4, pp. 77–83.
