Geodesic
On the structure of nets over quadratic fields
Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 3, pp. 87-95 Cet article a éte moissonné depuis la source Math-Net.Ru

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The structure of nets over quadratic fields is studied. Let $K=\mathbb{Q} (\sqrt{d})$ be a quadratic field, $\mathfrak{D}$ the ring of integers of the quadratic field $K$. A set of additive subgroups $\sigma=(\sigma_{ij})$, $1\leq i,j\leq n$, of a field $K$ is called a net of order $n$ over $K$ if $\sigma_{ir} \sigma_{rj} \subseteq{\sigma_{ij}} $ for all values of the index $i$, $r$, $j$. A net $\sigma=(\sigma_{ij})$ is called irreducible if all additive subgroups $\sigma_{ij}$ are different from zero. A net $\sigma = (\sigma_{ij})$ is called a $D$-net if $1 \in\tau_{ii}$, $1\leq i\leq n$. Let $\sigma = (\sigma_{ij})$ be an irreducible $D$-net of order $n\geq 2$ over $K$, where $\sigma_{ij}$ are $\mathfrak{D}$-modules. We prove that, up to conjugation diagonal matrix, all $\sigma_{ij}$ are fractional ideals of a fixed intermediate subring $P$, $\mathfrak{D}\subseteq P \subseteq K$, and all diagonal rings coincide with $P$: $\sigma_{11}=\sigma_{22}=\ldots =\sigma_{nn}=P,$ where $\sigma_{ij}\subseteq P$ are integer ideals of the ring $P$ for any $i, if $i>j$, then $P\subseteq\sigma_{ij}$. For any $i$, $j$ we have $\sigma_{1j}\subseteq\sigma_{ij}$. 
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S. S. Ikaev; V. A. Koibaev; A. O. Likhacheva. On the structure of nets over quadratic fields. Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 3, pp. 87-95. http://geodesic.mathdoc.fr/item/VMJ_2022_24_3_a6/

[1] Koibaev V. A., “O stroenii elementarnykh setei nad kvadratichnymi polyami”, Vladikavk. mat. zhurn., 22:4 (2020), 87–91 | DOI

[2] Borevich Z. I., Shafarevich I. R., Teoriya chisel, Nauka, M., 1985

[3] Dryaeva R. Yu., Koibaev V. A., Nuzhin Ya. N., “Polnye i elementarnye seti nad polem chastnykh koltsa glavnykh idealov”, Zap. nauch. seminarov POMI RAN, 455, 2017, 42–51

[4] Atya M., Makdonald I., Vvedenie v kommutativnuyu algebru, Mir, M., 1972

[5] Borevich Z. I., “O podgruppakh lineinykh grupp, bogatykh transvektsiyami”, Zap. nauch. seminarov LOMI, 75, 1978, 22–31

[6] Levchuk V. M., “Zamechanie k teoreme L. Diksona”, Algebra i logika, 22:4 (1983), 421–434

[7] Gilmer R., Ohm J., “Integral domains with quotient overrings”, Math. Ann., 153:2 (1964), 97–103