1. ## Ideals

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.

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!

2. Here's some useful theorems:

$\displaystyle I \subseteq R$ is a prime ideal $\displaystyle \iff R/I$ is an integral domain.

$\displaystyle I \subseteq R$ is a maximal ideal $\displaystyle \iff R/I$ is a field.

3. 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.

4. Originally Posted by brisbane
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.
$\displaystyle \mathbb{Z}[x]/(x) = \mathbb{Z}$, $\displaystyle \mathbb{Z}[x]/(3) = \mathbb{Z}_3[x]$, and $\displaystyle \mathbb{Z}[x]/(3,x) = \mathbb{Z}_3$

5. For the PID part, you need to check if every ideal can be written in the form $\displaystyle (p(x))$ for some $\displaystyle p(x)\in \mathbb{Z}[x]$. Maybe you can see if this is possible with the ideals you have. Can you find $\displaystyle p(x)$ such that $\displaystyle (p(x))=(3,x)$?

6. In a PID, every non zero prime ideal is also maximal (ask if you'd like to see why). For $\displaystyle \mathbb{Z}[x]$ consider $\displaystyle (x)$. What does this theorem tell us?