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...


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

Hint was to use \psi : R \rightarrow S/B defined by \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 Ker \psi but for whatever reason, it doesn't get me there... I think I need to figure out how to show that R/\phi^{-1}(B) is isomorphic to S/B which is a field... HELP PLEASE