Research Papers

Full and elementary nets over the field of fractions of a Dedekind domain

V. A. Koibaev<sup>ab

a</sup> North Ossetian State University after Kosta Levanovich Khetagurov
^b Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences

Abstract: The set $\sigma=(\sigma_{ij}), 1\leq{i, j}\leq{n},$ of additive subgroups $\sigma_{ij}$ of the field $K$ is called a net (carpet) over $K$ of order $n$ if $\sigma_{ir} \sigma_{rj} \subseteq{\sigma_{ij}}$ for all values of the indices $i, r, j.$ A net considered without a diagonal is called an elementary net. Based on the elementary net $\sigma$, an elementary net subgroup $E(\sigma)$ is determined, which is generated by elementary transvections $t_{ij}(\alpha) = e+\alpha e_{ij}$. An elementary net $\sigma$ is called closed if the elementary net subgroup $E(\sigma)$ does not contain new elementary transvections. Let $R$ be a Dedekind domain, $K$ be the field of fractions of the ring $R$, $\sigma=(\sigma_ {ij})$ be a complete (elementary) net of order $n\geq 2$ (respectively $n\geq 3$) over $K$, where the additive subgroups $\sigma_{ij}$ are non-zero $R$-modules. It is proved that, up to conjugation by a diagonal matrix, all $\sigma_{ij}$ are fractional ideals of a fixed intermediate subring $P$, $R\subseteq P \subseteq K$, and for all $i<j$ inclusions are performed $\pi_{ij}\pi_{ji}\subseteq P, \ \pi_{ij}\subseteq P\subseteq \pi_{j i}$. In particular, the elementary net $\sigma$ is closed.

Keywords: general and special linear groups, full and elementary nets (carpets) of additive subgroups, net subgroup.

Received: 20.12.2024