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stable general linear group

Let $R$ be an associative ring with 1. As usual ${\mathrm{GL}}_n(R)$ will denote the general linear group of $n\times n$ non-singular matrices over $R$. There is an embedding of ${\mathrm{GL}}_n(R)$ into ${\mathrm{GL}}_{n+1}(R)$ 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 $m_{n+1,n+1}^{\prime }=1$. This gives a nested sequence of groups

$${\mathrm{GL}}_1(R)\subset {\mathrm{GL}}_2(R)\subset \dots \subset {\mathrm{GL}}_n(R)\subset {\mathrm{GL}}_{n+1}(R)\subset \dots$$GL_1(R)\subset GL_2(R)\subset \ldots \subset GL_n(R)\subset GL_{n+1}(R)\subset \ldots

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