Overgroups of $E(m,R)\otimes E(n,R)$

In the present paper we study subgroups $E(m,R)\otimes E(n,R)\le H\le G=\operatorname{GL}(mn,R)$, under assumption that the ring $R$ is commutative, and $m,n\ge3$. We define the group $\operatorname{GL}_m\otimes\operatorname{GL}_n$ by equations, calculate the normaliser of the group $E(m,R)\otimes E(n,R)$ and associate to each intermediate subgroup $H$ a uniquely determined lower level $(A,B,C)$, where $A,B,C$ are ideals in $R$ such that $mA,A^2\le B\le A$ and $nA,A^2\le C\le A$. Lower level specifies the largest elementary subgroup such that $E(m,n,R,A,B,C)\le H$. The standard answer to this problem asserts that $H$ is contained in the normaliser $N_G(E(m,n,R,A,B,C))$. Bibl. – 46 titles.