On the structure of elementary nets over quadratic fields

V. A. Koibaev

Vladikavkazskij matematičeskij žurnal, Tome 22 (2020) no. 4, pp. 87-91

Résumé

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 $P$.