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 be rings and a homomorphism. If B is a maximal ideal in prove that is maximal in .

Hint was to use defined by

I think that it is the inverse part that is killing me. I know that in this mapping, B is the but for whatever reason, it doesn't get me there... I think I need to figure out how to show that is isomorphic to S/B which is a field... HELP PLEASE