Example of a commutative ring that is Artinian but not Noetherian 
I want to give an example of a commutative ring that is Artinian but not Noetherian. Is there any examples not very difficult?

I considered the ring $\mathbb{Z}+x\mathbb{Q}[x]$. It is not Noetherian. But can we prove it is Artinian or not?
 A: Consider the abelian group $R=\mathbb Z_{p^\infty }$. We define $xy=0$ for every $x,y\in R$. 
Now, it is not hard to see that $R$ is an artinian ring but $R$ is not a noetherian ring.
Note that there is a prominent theorem telling about the relationship between artinian rings and noetherian rings.

Akizuki–Hopkins–Levitzki theorem:
  Let $R$ be a unitary ring. If $R$ is a left (right) artinian ring, then  $R$ is automatically a left (right) noetherian ring.

A: The ring $\mathbb{Z}+x\mathbb{Q}[x]$ is not artinian since artinian integral domains are fields. (For example, in this ring $x$ is not invertible.)
A: The Hopkins-Levitzki theorem has already been given to answer the first question (and I can't really improve on that!), and someone has provided a reason your example ring is not Artinian. I'm offering a second way to see why your ring is not Artinian.
Your ring has an ideal $I=x\Bbb Q[x]$, and $R/I\cong\Bbb Z$. If $R$ were Artinian, its quotients would all also be Artinian, but hopefully you know that $\Bbb Z$ is not Artinian. 
