On the normal structure of the general linear group over a ring

The article is devoted to investigation of the normal subgroups of the general linear group over a ring and centrality of the extension $\mathrm{St}(n,R)\to E(n,R)$. The notions of the standard commutator formula and the standard normal structure of $\mathrm{GL}(n,R)$, $\mathrm{E}(n,R)$, and $\mathrm{St}(n,R)$ and their relationships are discussed. In particular, it is shown that the normality of $\mathrm{E}(n,R)$ in $\mathrm{GL}(n,R)$ and the standard distribution of subgroups normalized by $\mathrm{E}(n,R)$ follow from some conditions of linear dependence in $R$. Also, it is proved that the standardness of the normal structure of $\mathrm{GL}(n,R)$ and centrality of $K_2(n,R)$ in $\mathrm{St}(n,R)$ follow from the same conditions over a quotient ring $R/I$ provided $\mathrm{sr}I\le n-1$. Under some additional assumptions (e.g. $I$ is contained in the Jacobson radical of $R$) the converse is also true. The standard tecnique due to H. Bass, Z. I. Borevich, N. A. Vavilov, L. N. Vaserstein, W. van der Kallen, A. A. Suslin, M. S. Tulenbaev, and others is used and developed in the article.