Isomorphisms of the general linear group over an associative ring

It is proved that every isomorphism of linear groups $\varphi\colon\mathrm{GL}_n(R)\to\mathrm{GL}_m(S)$ over arbitrary associative rings $R$ and $S$ with $1/2\in R$ and $1/2\in S$ for $n,m\ge3$ is a standard one on a subgroup $\mathrm{GE}_n(R)$ generated by elementary and diagonal matrices.