Full and elementary nets over the field of fractions of a ring with the QR-property

R. Yu. Dryaeva   ; V. A. Koibaev

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 30 (2024) no. 4, pp. 77-83

Résumé

The set $\sigma=(\sigma_{ij})$, $1\leq{i, j}\leq{n},$ of additive subgroups $\sigma_{ij}$ of a 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 $i$, $r$, and $j$. A net considered without the diagonal is called an elementary net. Based on an elementary net $\sigma$, an elementary net subgroup $E(\sigma)$ is defined, which is generated by elementary transvections $t_{ij}(\alpha) = e+\alpha e_{ij}$. An elementary net $\sigma$ is called closed if the subgroup $E(\sigma)$ does not contain new elementary transvections. Suppose that $R$ is a Noetherian domain with the QR-property (i.e., any intermediate subring lying between $R$ and its field of fractions $K$ is a ring of fractions of the ring $R$ with respect to a multiplicative system in $R$), $\sigma=(\sigma_{ij})$ is a complete (elementary) net of order $n\geq 2$ ($n\geq 3$, respectively) over $K$, and the additive subgroups $\sigma_{ij}$ are nonzero $R$-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 the inclusions $\pi_{ij}\pi_{ji}\subseteq P$ and $\pi_{ij}\subseteq P\subseteq\pi_{j i}$ hold for all $i$. In particular, the elementary net $\sigma$ is closed.

Détail

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

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