The proofs I have seen of "N is a maximal ideal in a ring R iff R/N is a simple ring" involve a step using the canonical homomorphism $\phi$ and an ideal of $R/N$.
For example, suppose $\phi: R \rightarrow R/N$ is the canonical homomorphism and $J$ is an ideal of $R/N$. The next step is usually something like $N \subset \phi^{-1}(J) \subset R$.
I don't understand why $N$ is contained in $\phi^{-1}(J)$.
Everything in $N$ maps to zero under $\phi$, and the ideal $J$ contains zero, and so the preimage $\phi^{-1}(J)$ contains everything that maps to zero, so $\phi^{-1}(J)$ contains $N$.