Prime ideals of the matrix ring
You just need to convince yourself that $M_n(A)M_n(B)=M_n(AB)$ for any two ideals $A,B$ of $R$.
After that it's just straightforward checking:
If $P$ is prime, then what does $M_n(A)M_n(B)\subseteq M_n(P)$ imply about $M_n(A)$ or $M_n(B)$?
If $M_n(P)$ is prime (and you don't know if $P$ is yet), then take two ideals $A,B$ and suppose $AB\subseteq P$ and ask what that means for $M_n(A)$ and $M_n(B)$. Let me know if you get stuck.