On the structure of nets over quadratic fields

S. S. Ikaev ; V. A. Koibaev ; A. O. Likhacheva

Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 3, pp. 87-95

Résumé

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 $\sigma_{1j}\subseteq\sigma_{ij}$.