Here is the problem I have:

Let be a PID and . Prove that the ideal is maximal in if and only if is a prime element in . (Recall is a prime in if is not a unit and if then or )

I know that prime elements generate prime ideals (and I think I can prove that if I have to) and the only trick I had up my sleeve was that prime ideal -> ID and finite ID's are fields -> maximal, but it's not finite, so I am having difficulties constructing a proof...(Doh)