Ideals

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May 8th 2010, 07:00 PM
brisbane

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!
May 8th 2010, 07:16 PM
chiph588@

Here's some useful theorems:

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

$I \subseteq R$ is a maximal ideal $\iff R/I$ is a field.
May 8th 2010, 07:35 PM
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.
May 8th 2010, 07:44 PM
chiph588@

Quote:

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.

$\mathbb{Z}[x]/(x) = \mathbb{Z}$ , $\mathbb{Z}[x]/(3) = \mathbb{Z}_3[x]$ , and $\mathbb{Z}[x]/(3,x) = \mathbb{Z}_3$
May 8th 2010, 07:52 PM
roninpro

For the PID part, you need to check if every ideal can be written in the form $(p(x))$ for some $p(x)\in \mathbb{Z}[x]$ . Maybe you can see if this is possible with the ideals you have. Can you find $p(x)$ such that $(p(x))=(3,x)$ ?
May 8th 2010, 08:09 PM
chiph588@

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