For example:

Prove (3,x) is maximal in Z[x]. Is (2,x) maximal? (5,x)?

Prove that (3) and (x) are prime ideals in Z[x].

Is Z[x] a PID? I understand why Z is a PID but can't generalize past it.

If anyone can also show me some more esoteric examples (since these are pretty standard ones), that'd be great. Thanks!