nLab stable general linear group

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

$$GL_1(R)\subset GL_2(R)\subset \ldots \subset GL_n(R)\subset GL_{n+1}(R)\subset \ldots$$

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