Suppose $A,\,B$ are (possibly noncommutative) rings, and $GL_n(-)$ is the group of invertible $n\times n$ matrices over a given ring. Suppose $f:A\to B$ is surjective, does it necessarily follow that $f_\ast:GL_n(A)\to GL_n(B)$ is surjective for $n>1$? If not, why not?
I know that this need not be true for $n=1$, but is true for the subgroup $E_n(-)$ generated by elementary matrices. If it isn't true in general, are there any conditions upon the rings or homomorphism that ensures this is true?