Structure of Nets over Quadratic Fields

Abstract

We study the structure of nets over quadratic fields. Let $K=𝕈(\sqrt{d}\,)$ be a quadratic field, and let $\mathfrak{D}$ be the ring of integers of $K$. A set $\sigma=(\sigma_{ij})$ of additive subgroups of $K$ is a net (carpet) of order $n$ over $K$ if $\sigma_{ir}\sigma_{rj}\subseteq{\sigma_{ij}}$ for all values of the indices $i$, $r$, $j$, ${1\leq i,r,j\leq n}$. A net $\sigma=(\sigma_{ij})$ is irreducible if all additive subgroups $\sigma_{ij}$ are nonzero. A net $\sigma=(\sigma_{ij})$ is a $D$-net if $1\in\sigma_{ii}$, $1\leq i\leq n$. 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 by a 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 $P$; i.e., $\sigma_{11}=\sigma_{22}=\dots=\sigma_{nn}=P$, where $\sigma_{ij}\subseteq P$ are integer ideals of $P$ for all $i<j$, and $P\subseteq\sigma_{ij}$ if $i>j$. Furthermore, $\sigma_{1j}\subseteq\sigma_{ij}$ for all $i$ and $j$.