Ideals

Apr 2009
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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!
 

chiph588@

MHF Hall of Honor
Sep 2008
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Champaign, Illinois
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.
 
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Apr 2009
18
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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.
 

chiph588@

MHF Hall of Honor
Sep 2008
1,163
429
Champaign, Illinois
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 \(\displaystyle \mathbb{Z}[x]/(3,x) = \mathbb{Z}_3 \)
 
Nov 2009
485
184
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 \(\displaystyle (p(x))=(3,x)\)?
 

chiph588@

MHF Hall of Honor
Sep 2008
1,163
429
Champaign, Illinois
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?
 
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