Stable general linear group

Remarks

Equivalently, $GL(R)$ is the group of infinite matrices $(a_{ij})_{i, j \ge 1}$ over $R$ that are invertible and differ from the identity matrix in only finitely many entries. Each $GL_n(R)$ is a subgroup of $GL(R)$, and $GL(R)$ is their union. Example: over $\Z$, the matrices $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ and $\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ give the same element of $GL(\Z)$, since the second is $\iota_2$ of the first. Non-example: the infinite diagonal matrix with every diagonal entry equal to $-1$ is not in $GL(\Z)$, since it differs from the identity in infinitely many entries.