Net rings normalized by a nonsplit maximal torus

We investigate net rings $M(\sigma)$ normalized by a torus $T=T(d)$, which is the image of the multiplicative group of the radical extension $K=k(\sqrt[n]d)$ (of degree $n$ of a field $k$, $char(k)\neq2$) under the regular embedding into $G=GL(n,k)$. It is shown that the structure of these net rings is determined by a certain subring of the ground field $k$. Necessary and sufficient conditions are obtained for the normalizability of a net ring $M(\sigma)$ by the torus $T=T(d)$ for the case when the ground field $k=\mathbb Q$ is the field of rational numbers. We also study transvection modules and factor rings of intermediate subgroups $H$, $T\subseteq H\subseteq G$.