Check Noetherian-Artinian for $\mathbb{Q}[x]$ as a $\mathbb{Q}$-module and $\mathbb{Q}[x]$-module

Question

I have to check if these modules are Artinian or/and Noetherian.

$\mathbb{Q}[x]$ as a $\mathbb{Q}$-module

$\mathbb{Q}[x]$ as a $\mathbb{Q}[x]$-module

For the second one I know that $\mathbb{Q}$ is a field so it's Noetherian, so $\mathbb{Q}[x]$ is also Noetherian. Furthermore, being Noetherian as a ring means you are Noetherian as a module over yourself, so $\mathbb{Q}[x]$ is Noetherian as a $\mathbb{Q}[x]$-module.

Also, $(x) \supseteq (x^{2}) \supseteq (x^{3}) \supseteq \cdots $ is an infinite decreasing sequence of ideals. So it's not Artinian.

Am I right?

And what about the first one? I think it's neither Artinian nor Noetherian but how can I say it?

Answer 1

Your answer for $\mathbb{Q}[x]$ is correct.

Hint: is $x^2$ in the module generated by $x$ when the ring is $\mathbb{Q}$?

Answer 2

You have an infinite increasing sequence of $\mathbb{Q}$-vector spaces $\mathbb{Q}\subset \mathbb{Q}+x\mathbb{Q}\subset \cdots$, so...

You also have an infinite deacreading sequence of $\mathbb{Q}$-vector spaces $\cdots x^2\mathbb{Q}[x]\subset x\mathbb{Q}[x]\subset \mathbb{Q}[x]$