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 Vladikavkaz. Mat. Zh., 2020, Volume 22, Number 4, Pages 87–91 (Mi vmj746)

On the structure of elementary nets over quadratic fields

V. A. Koibaevab

a Southern Mathematical Institute VSC RAS, 22 Markus St., Vladikavkaz 362027, Russia
b North-Ossetian State University after K. L. Khetagurov, 44 Vatutina St., Vladikavkaz 362025, Russia

Abstract: The structure of elementary nets over quadratic fields is studied. A set of additive subgroups $\sigma=(\sigma_{ij})$, $1\leq i,j\leq n$, of a ring $R$ is called a net of order $n$ over $R$ if $\sigma_{ir} \sigma_{rj} \subseteq{\sigma_{ij}}$ for all $i$, $r$, $j$. The same system, but without the diagonal, is called elementary net (elementary carpet). An elementary net $\sigma=(\sigma_{ij})$ is called irreducible if all additive subgroups $\sigma_{ij}$ are different from zero. Let $K=\mathbb{Q} (\sqrt{d} )$ be a quadratic field, $D$ a ring of integers of the quadratic field $K$, $\sigma = (\sigma_{ij})$ an irreducible elementary net of order $n\geq 3$ over $K$, and $\sigma_{ij}$ a $D$-modules. If the integer $d$ takes one of the following values (22 fields): $-1$, $-2$, $-3$, $-7$, $-11$, $-19$, $2$, $3$, $5$, $6$, $7$, $11$, $13$, $17$, $19$, $21$, $29$, $33$, $37$, $41$, $57$, $73$, then for some intermediate subring $P$, $D\subseteq P\subseteq K$, the net $\sigma$ is conjugated by a diagonal matrix of $D(n, K)$ with an elementary net of ideals of the ring $P$.

Key words: net, carpet, elementary net, closed net, algebraic number field, quadratic field.

DOI: https://doi.org/10.46698/h3104-8810-6070-x

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UDC: 512.5
MSC: 20G15

Citation: V. A. Koibaev, “On the structure of elementary nets over quadratic fields”, Vladikavkaz. Mat. Zh., 22:4 (2020), 87–91

Citation in format AMSBIB
\Bibitem{Koi20} \by V.~A.~Koibaev \paper On the structure of elementary nets over quadratic fields \jour Vladikavkaz. Mat. Zh. \yr 2020 \vol 22 \issue 4 \pages 87--91 \mathnet{http://mi.mathnet.ru/vmj746} \crossref{https://doi.org/10.46698/h3104-8810-6070-x}