# Ideals

#### brisbane

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
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.

• brisbane

#### 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.

#### chiph588@

MHF Hall of Honor
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 $$\displaystyle \mathbb{Z}[x]/(3,x) = \mathbb{Z}_3$$

#### roninpro

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
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?

• roninpro
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