Does a ring map $f:R\to S$ induce a homomorphism $GL_n(R)\to GL_n(S)$?

Question

Let $R$ and $S$ be commutative rings with $1$ and $f:R\to S$ a ring homomorphism. Does $f$ induce a group homomorphism $GL_n(R) \to GL_n(S)$?

Progress

I first consider the map $\bar{f}:M_n(R)\to M_n(S)$ given by letting $f$ act on the matrix elements. Using the formula for the determinant, I obtain $\det(\bar{f}(A))=f(\det(A))$. Am I on the right track?

Using Jyrki Lahtonen's hint, I now see that if $A$ is invertible then $f(\det(A))$ is a unit of $S$. So $\bar{f}$ maps $GL_n(R)$ into $GL_n(S)$.