# nLab stable general linear group

Let $R$ be an associative ring with 1. As usual ${\mathrm{GL}}_{n}\left(R\right)$ will denote the general linear group of $n×n$ non-singular matrices over $R$. There is an embedding of ${\mathrm{GL}}_{n}\left(R\right)$ into ${\mathrm{GL}}_{n+1}\left(R\right)$ sending a matrix $M=\left({m}_{i,j}\right)$ to the matrix ${M}^{\prime }$ obtained from $M$ by adding an extra row and column of zeros except that ${m}_{n+1,n+1}^{\prime }=1$. This gives a nested sequence of groups

${\mathrm{GL}}_{1}\left(R\right)\subset {\mathrm{GL}}_{2}\left(R\right)\subset \dots \subset {\mathrm{GL}}_{n}\left(R\right)\subset {\mathrm{GL}}_{n+1}\left(R\right)\subset \dots$GL_1(R)\subset GL_2(R)\subset \ldots \subset GL_n(R)\subset GL_{n+1}(R)\subset \ldots

and we write $\mathrm{GL}\left(R\right)$ for the colimit (union in this case) of these. It will be called the stable general linear group over $R$.