When does a ring map $R\to S$ produce a group epimorphism $GL_n(R)\to GL_n(S)$? Let $R$ and $S$ be rings with $1$ (not necessarily commutative) and $f:R\to S$ a ring homomorphism preserving $1$. Let $\bar{f}$ be the ring map $M_n(R)\to M_n(S)$ given by  $f$ acting on the matrix elements. 
My question is, what is the most general condition on $f$ for $\bar{f}$ to be a surjective group homomorphism $GL_n(R)\twoheadrightarrow GL_n(S)$? I think this is true if $f$ is an isomorphism but I can't be sure.
Many thanks for your help.

Here are some standard examples: Consider $a$, $b$ integers $>0$ so that $b \mid a \ $. We have the canonical surjective morphism of rings 
$$\mathbb{Z}/(a) \to \mathbb{Z}/(b)$$
and the induced morphism of groups 
$$GL_n(\mathbb{Z}/(a)) \to GL_n(\mathbb{Z}/(b))$$
is surjective. 
Even for $n=1$ this is a good exercise. 
The general case follows from the surjectivity of  
$$SL_n(\mathbb{Z}) \to SL_n(\mathbb{Z}/(b))$$
and the $GL_1 (\cdot )$ case. 
Also see http://en.wikipedia.org/wiki/Approximation_in_algebraic_groups
