I'm wondering how to go about proving that certain ideals are principal or prime or maximal. I don't have a grasp of the general method if I'm given an ideal in a ring and told to prove it is maximal or prime, or if I'm given a ring and told to prove that it is (or isn't) a PID.
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!
Here's some useful theorems:
is a prime ideal is an integral domain.
is a maximal ideal is a field.
Thanks, but I know those and need a little more handholding than that. I'm missing being able to do the step where I show Z[x]/3 or Z[x]/x is an integral domain, or that Z[x]/(3,x) is a field.
For the PID part, you need to check if every ideal can be written in the form for some . Maybe you can see if this is possible with the ideals you have. Can you find such that ?
In a PID, every non zero prime ideal is also maximal (ask if you'd like to see why). For consider . What does this theorem tell us?