Ring homomorphisms with Matrix Ring domains

Question

If $M(R)$ is the full ring of $n\times n$ matrices with entries from a ring $R$ and $f\colon M(R) \to M(S)$ is a ring homomorphism, is it true that $f$ is actually a map of the form $M(g)$ for some ring homomorphism $g\colon R\to S$? Why?

Answer 1

Let $n=2$ and consider the ring homomorphism $$\pmatrix{a&b\cr c&d\cr}\mapsto \pmatrix{a&-b\cr-c&d\cr}$$