I missed this problem on a midterm and I cannot figure out how to use the F.I.T. to show this... if anyone can help I would really appreciate it...

Problem:

Let$\displaystyle R,S$ be rings and $\displaystyle \phi :R \rightarrow S$ a homomorphism. If B is a maximal ideal in $\displaystyle S$ prove that $\displaystyle \phi^{-1}(B)$ is maximal in $\displaystyle R$.

Hint was to use $\displaystyle \psi : R \rightarrow S/B$ defined by $\displaystyle \phi(r) = r+B$

I think that it is the inverse part that is killing me. I know that in this mapping, B is the $\displaystyle Ker \psi$ but for whatever reason, it doesn't get me there... I think I need to figure out how to show that $\displaystyle R/\phi^{-1}(B)$ is isomorphic to S/B which is a field... HELP PLEASE