As the titles says, I need to show that in a PID $R$ an ideal is maximal iff it is prime. This is easy to do if $R$ has a multiplicative identity. I can not do it if $R$ does not have an identity. It would be great if someone could help me out.
This is a CW answer intended to help clear this question from the unanswered queue.
It's really unlikely Hungerford intended to ask about a domain without a unit. For instance, you can see that he assumes integral domains have a unit in the proof to theorem 3.4. If he did not intend for there to be units, he would have a really hard time defining irreducible elements and handling UFD's to begin with. Finally, most authors don't put unnecessarily complicated problems as problem #1 in a section.