principal ideals, integral domains, ideals,?

Question

I am stuck trying to grasp this concept. I know that

1. $\Bbb{Z}$ is a PID,

2. $R=\Bbb{Z}[X]$ is not a PID,

3. $\Bbb{Z}[i]$ is a PID.

If someone could help me grasp these concepts it would be helpful. Ideally I would like an example of each please?

This is my thinking:

1. If you have $\Bbb{Z}_6$, this is not a PID because it isnt a ID as $2*3=3*2$? But $\Bbb{Z}_7$ is a PID as it is a ID.
2. I am struggling to grasp this one espically, e.g. $\Bbb{Z}[5]$ or $\Bbb{Z}[3]$.
3. As there isn't any unique decomposition meaning like $2*3=3*2$ every $\Bbb{Z}[i]$ is a PID? But then I don't understand why $\Bbb{Z}[\sqrt{-5}]$ is not a PID?

Is my thinking correct? Because I am getting mixed up.

Thanks!

Answer 1

I guess I will just write up the answer as to why $\mathbb{Z}[\sqrt{5}]$ is not a PID.

Clearly it suffices to find an ideal which is not principal, ie an ideal which can't possibly be generated by one element. As your guess ($2, \sqrt{5}$) works but you can't just state it, you actually have to prove it.

Suppose it can be generated by element $\alpha \in \mathbb{Z}[\sqrt{5}]$. Then $2 = \alpha \times \beta$ and $\sqrt{5} = \alpha \times \gamma$ for some $\beta, \gamma \in \mathbb{Z}[\sqrt{5}]$. Can you prove such $\alpha$ can not exist?