The net and elementary net group associated with non-split maximal torus

N. A. Djusoeva

Vladikavkazskij matematičeskij žurnal, Tome 17 (2015) no. 2, pp. 12-15

Résumé

The elements of matrixes of a non-split maximal torus $T=T(d)$ (associated with a radical extension $k(\sqrt[n]d)$ of degree $n$ of the ground field $k$) generate some subring $R(d)$ of the field $k$. Let $R$ be an intermediate subring, $R(d)\subseteq R\subseteq k$, $d\in R$, $A_1\subseteq\dots\subseteq A_n$ be a chain of ideals of the ring $R$, and $dA_n\subseteq A_1$. By $\sigma=(\sigma_{ij})$ we denote the net of ideals defined by $\sigma_{ij}=A_{i+1-j}$ with $j\leq i$ and $\sigma_{ij}=dA_{n+i+1-j}$ with $j\geq i+1$. By $G(\sigma)$ and $E(\sigma)$ we denote the net and the elementary net group, respectively. It is proved, that $TG(\sigma)$ and $TE(\sigma)$ are intermediate subgroups of $GL(n, k)$ containing the torus $T$.