How to prove this conclusion? I don't know if it is related to ring homomorphism.

Question

Assuming $F$ is a number field, and $M_n(F)$ represents the set of all $n\times n$ matrixes on the number field $F$, $M_m(F)$ is defined similarly. Map $f:M_n(F)\to M_m(F)$ meets conditions below: 1) $f$ is injection, 2) $f(A+B)=f(A)+f(B)$, 3) $f(AB)=f(A)f(B)$, 4)$f(I_n)=I_m,f(0_n)=0_m$.Then how do we prove that $n| m$? One thing is, I not sure how to solve it without using any abstact algebra knowledge,becuase this is suppose to be a test in linear algebra.