General linear group over a local ring

Definition
Let $$R$$ be a local ring with unity, viz a ring which has a multiplicative identity and also has a unique maximal ideal. Then, for any positive integer $$n$$ we define the general linear group of order $$n$$ over $$R$$, denoted $$GL(n,R)$$, as the group of all invertible matrices of order $$n$$ with entries in $$R$$.

Relation with the general linear group over the residue field
Let $$R$$ be a local ring with unique maximal ideal $$M$$ and residue field $$k$$ (in other words $$k$$ is the quotient of $$R$$ by $$M$$). Then, there is a natural map from the matrix ring over $$R$$ to the matrix ring over $$k$$ that sends each element to its congruence class modulo $$M$$. It can be seen that $$GL(n,R)$$ is precisely equal to the inverse image of $$GL(n,k)$$ under this map.